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isIsomorphic -- probabilistic test for isomorphism of modules

Description

In case both modules are homogeneous the program first uses checkDegrees to see whether an isomorphism is possible. This may be an isomorphism up to shift if Strict => false (the default) or on the nose if Strict => true.

i1 : S = ZZ/32003[x_0..x_3]

o1 = S

o1 : PolynomialRing
 
i2 : m = random(S^3, S^{4:-2});

             3      4
o2 : Matrix S  <-- S
 
i3 : M = coker m;
 
i4 : assert isIsomorphic(S^{-3} ** M, M)
 
i5 : assert not isIsomorphic(S^{-3} ** M, M, Strict => true)

If this test is passed, the program uses a variant of the Hom command to compute a random map of minimal possible degree from $M$ to $N$, and checks whether this is surjective and injective. If Tries => n is provided, this is attempted $n$ times, otherwise a heuristic based on the characteristic of the field is used to determine the number of attempts to make (1 in characteristic zero, ~10 in characteristic 32003, and ~150 in characteristic 2).

i6 : A = random(target m, target m);

             3      3
o6 : Matrix S  <-- S
 
i7 : B = random(source m, source m);

             4      4
o7 : Matrix S  <-- S
 
i8 : N = coker(A * m * B);
 
i9 : isIsomorphic(N, M)

o9 = true
 
i10 : isomorphism(N, M)

o10 = | 1     140  14512 |
      | 2017  6517 3768  |
      | 13952 1685 6668  |

o10 : Matrix N <-- M

The internal routine for generating a random map of minimal possible degree is essentially equivalent to the code below, but caching and various other optimizations make this routine more efficient.

i11 : f = homomorphism random Hom(N, M, DegreeLimit => 0)

o11 = | 1093 9784   -5815  |
      | 9234 -14769 -2171  |
      | 8559 -14316 -14077 |

o11 : Matrix M <-- N
 
i12 : isIsomorphism f

o12 = true

If the modules are inhomogeneous and Homogeneous => false is passed, the random map is a random linear combination of the generators of the module of homomorphisms.

If the output is false, then the conclusion of non-isomorphism is only probabilistic. If the output is true, then as certificate an isomorphism $M \to N$ is cached in the youngest module and can be retried using the isomorphism method.

The following examples checks two well-known isomorphisms in homological algebra.

i13 : S = ZZ/32003[x_0..x_3]

o13 = S

o13 : PolynomialRing
 
i14 : I = monomialCurveIdeal(S, {1,3,5})

              2          2      2     3    2
o14 = ideal (x  - x x , x x  - x x , x  - x x )
              2    1 3   1 2    0 3   1    0 2

o14 : Ideal of S
 
i15 : c = codim I

o15 = 2
 
i16 : W = Ext^c(S^1/I, S^1)

o16 = cokernel {-4} | x_2 x_1 x_0^2 |
               {-4} | x_3 x_2 x_1^2 |

                             2
o16 : S-module, quotient of S
 
i17 : H = Hom(S^1/I, S^1/(I_0,I_1))

o17 = subquotient (| x_3 x_2 |, | x_2^2-x_1x_3 x_1^2x_2-x_0^2x_3 |)

                                1
o17 : S-module, subquotient of S
 
i18 : isIsomorphic(W, H)

o18 = true
 
i19 : isomorphism(W, H)

o19 = {-4} | 1 0  |
      {-4} | 0 -1 |

o19 : Matrix W <-- H
 
i20 : mm = ideal gens S

o20 = ideal (x , x , x , x )
              0   1   2   3

o20 : Ideal of S
 
i21 : T1 = Tor_1(W, S^1/(mm^3))

o21 = subquotient ({-3} | x_3^2 x_2x_3 x_1x_3 x_0x_3 x_2^2 x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 0     0      0      0      0     0      0      0     0      0     0   0   0   0   |, {-3} | 0            x_3^3 x_2x_3^2 x_1x_3^2 x_0x_3^2 x_2^2x_3 x_1x_2x_3 x_0x_2x_3 x_1^2x_3 x_0x_1x_3 x_0^2x_3 x_2^3 x_1x_2^2 x_0x_2^2 x_1^2x_2 x_0x_1x_2 x_0^2x_2 x_1^3 x_0x_1^2 x_0^2x_1 x_0^3 0     0        0        0        0        0         0         0        0         0        0     0        0        0        0         0        0     0        0        0     0     0        0        0        0        0         0         0        0         0        0     0        0        0        0         0        0     0        0        0     |)
                   {-3} | 0     0      0      0      0     0      0      0     0      0     x_3^2 x_2x_3 x_1x_3 x_0x_3 x_2^2 x_1x_2 x_0x_2 x_1^2 x_0x_1 x_0^2 0   0   0   0   |  {-3} | 0            0     0        0        0        0        0         0         0        0         0        0     0        0        0        0         0        0     0        0        0     x_3^3 x_2x_3^2 x_1x_3^2 x_0x_3^2 x_2^2x_3 x_1x_2x_3 x_0x_2x_3 x_1^2x_3 x_0x_1x_3 x_0^2x_3 x_2^3 x_1x_2^2 x_0x_2^2 x_1^2x_2 x_0x_1x_2 x_0^2x_2 x_1^3 x_0x_1^2 x_0^2x_1 x_0^3 0     0        0        0        0        0         0         0        0         0        0     0        0        0        0         0        0     0        0        0     |
                   {-2} | 0     0      0      0      0     0      0      0     0      0     0     0      0      0      0     0      0      0     0      0     x_3 x_2 x_1 x_0 |  {-2} | x_2^2-x_1x_3 0     0        0        0        0        0         0         0        0         0        0     0        0        0        0         0        0     0        0        0     0     0        0        0        0        0         0         0        0         0        0     0        0        0        0         0        0     0        0        0     x_3^3 x_2x_3^2 x_1x_3^2 x_0x_3^2 x_2^2x_3 x_1x_2x_3 x_0x_2x_3 x_1^2x_3 x_0x_1x_3 x_0^2x_3 x_2^3 x_1x_2^2 x_0x_2^2 x_1^2x_2 x_0x_1x_2 x_0^2x_2 x_1^3 x_0x_1^2 x_0^2x_1 x_0^3 |

                                3
o21 : S-module, subquotient of S
 
i22 : T2 = Tor_1(S^1/(mm^3), W)

o22 = subquotient ({-1} | 1 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 x_2 x_1 0   0   0   x_0 0    -x_3 0   0   0    0   0   0    -x_3 0   0    0    -x_3 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |, {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    -x_3 0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |)
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 x_3 x_2 x_2 0   0   0   -x_1 0    x_3 0   0    0   x_3 0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | x_2 -x_1 0   0    0   0    x_3 -x_2 0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    -x_3 0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 1 0  1 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   x_0 x_2 x_1 0   0    0    0   0   -x_3 0   0   0    0    0   0    -x_3 0    0    0    -x_3 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | x_0 0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    -x_3 0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   x_3 x_2 x_1 x_0  0    0   x_2 0    x_3 0   0    0    x_3 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   x_0  x_2 -x_1 0   0    0   0    x_3  -x_2 0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    -x_3 0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 1 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    x_2  x_1 x_0 0    0   0   -x_3 0    0   0    0    0    0    -x_3 0    0    0    -x_3 0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    x_0 0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    -x_3 0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 1 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   x_0  x_2 -x_1 0   0    0    0    0   0    x_3  -x_2 0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    -x_3 0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 -1 0 0 1 0 0 0 0  1 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   x_2  x_1 x_0 0    0    0   -x_3 0    0    0    0    0    0    -x_3 0    0    0    0    0    0    -x_3 0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    x_0 0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    -x_3 0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 1 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   x_0  0   0    0    0    0   0    0    0    0   0    x_3  -x_2 0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    -x_3 0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 1  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    x_0 0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    -x_3 0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  1 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   x_0  -x_2 x_1  x_2 -x_1 0    0    0   0    0    0    x_3 -x_2 0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    -x_3 0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 1 0 0 0  0 0  1 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   x_2  x_1  x_0 0    0    0    -x_3 0    0    0    0    0    0    0    -x_3 0    -x_3 0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    x_0 0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    -x_3 0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 1 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   x_0  -x_2 x_1  x_2 -x_1 0    0    0   0    x_3  -x_2 0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    -x_3 0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 1 0 -1 0 0  0 0  1 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   x_2  x_1  x_0  0    0    0    -x_3 0    0    0    -x_3 0    0    0    0    0    0    0    0    0    -x_3 0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    x_0 0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    -x_3 0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 1 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   x_0  -x_2 x_1  0   0    0    0    0   0    x_3  -x_2 0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    -x_3 0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 -1 0 0  0 1 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    x_2  x_1  x_0  0    0    0    0    0    0    -x_3 0    0    0    0    -x_3 0    -x_3 0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    x_0 0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    -x_3 0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 1 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   x_0  -x_2 x_1  x_2 -x_1 0    0    x_3 -x_2 0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    -x_3 0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 1 0  0 0  0 -1 0 0 0  1 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    x_2  x_1  x_0  -x_3 0    0    0    0    0    0    -x_3 0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    x_0 0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    -x_3 0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 1 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   x_0  -x_2 x_1  0   0    x_3  -x_2 0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    -x_3 0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 -1 0 1 0 0  1 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  0    0    0    -x_3 0    0    0    0    0    0    -x_3 0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    x_0 0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    -x_3 0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 1 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   x_0  -x_2 x_1  0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    -x_3 0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 1  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    x_0  0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    -x_3 0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  1 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    x_0  -x_2 x_1  x_2 -x_1 0    0    0   0    0    0    -x_3 x_2  x_3 -x_2 0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    -x_3 0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 1  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    x_0 0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    -x_3 0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  1 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   x_0  -x_2 x_1  x_2 -x_1 0    0    0    0    0   0    -x_3 x_2  x_3  -x_2 0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    -x_3 0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 1  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    x_0 0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    -x_3 0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  1 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   x_0  -x_2 x_1  0    0    0   0    0    0    0    0    0   0    -x_3 x_2  x_3  -x_2 0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    -x_3 0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 1 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  0    0    0    -x_3 0    0    0    0    -x_3 0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    x_0 0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    -x_3 0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 1 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   x_0  0    0    -x_2 x_1  x_2 -x_1 0    0    0    0    -x_3 x_2  x_3 -x_2 0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    -x_3 0   0    0    0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 1  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    x_0 0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    -x_3 0    0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  1 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   x_0  0    0    -x_2 x_1  0    0    0   0    -x_3 x_2  x_3  -x_2 0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    -x_3 0    0    0    0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 -1 0 1 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  0    0    0    -x_3 0    0    0    -x_3 0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    x_0 0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    -x_3 0    0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 1 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   x_0  0    0    -x_2 x_1  -x_3 x_2  0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    -x_3 0    0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  0    0    0    0    0    -x_3 |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    x_0  0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    -x_3 0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 1 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    x_0  -x_2 x_1  x_2 -x_1 0    0    -x_3 x_2  x_3 -x_2 0    0    0    0    0    0    0    -x_3 0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 1  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    x_0 0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    -x_3 0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  1 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   x_0  -x_2 x_1  0    0    0   0    -x_3 x_2  x_3  -x_2 0    0    0    0    0    -x_3 0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 1 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  -x_3 0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    x_0 0    0    0    0    0    0    0    0    0    0    0    -x_3 0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 1 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   x_0  0    0    -x_2 x_1  -x_3 x_2  0    0    0    0    0    -x_3 0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 0   0   0     |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 0 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_1  x_0  |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    x_0  0    0    0    0    0    0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_2 x_1 x_0^2 |
                   {-1} | 0 0 0  0 0 0 0 0 0 0  0 0  0 0  0 0 0  0 0 0 0  0 0 1 0   0   0   0   0   0   0    0    0   0   0    0   0   0    0    0   0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    0    |  {-1} | 0   0    0   0    0   0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0   0    0    0    0    0    0    0    0    0    0    0    0   0    0    0    0    0    0   0    0    0    0    0    0    0    0    x_0  -x_2 x_1  -x_3 x_2  0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     0   0   0     x_3 x_2 x_1^2 |

                                40
o22 : S-module, subquotient of S
 
i23 : elapsedTime isIsomorphic(T1, T2)
 -- 3.96203s elapsed

o23 = true
 
i24 : elapsedTime isomorphism(T1, T2)
 -- .000045355s elapsed

o24 = {-1} | 1      -3976  -13490 13495  -2886  2577   14757  -881   7677  
      {-1} | -2527  -13566 2778   -6934  -14806 4619   -13099 6022   -10907
      {-1} | -15420 5642   1489   1354   4591   11881  -5253  7296   -1098 
      {-1} | 7909   -12428 -2260  -8465  12113  -6893  8411   4186   -9393 
      {-1} | -9615  2934   10440  5015   8145   -5585  1360   3295   12851 
      {-1} | -4881  -7984  12700  -10391 -10009 -14538 13207  262    -6500 
      {-1} | -11807 -9742  -1522  -7079  9115   8111   -4535  -9862  -8598 
      {-1} | 1622   15897  13496  8610   3523   -15175 -1378  13625  -8157 
      {-1} | 1605   -3914  -9771  3286   -13647 -9975  5030   -3778  7483  
      {-1} | 3413   -8536  -2477  -14194 -3535  4490   15882  -4198  -13581
      {-1} | 7714   10139  3990   8809   -13687 -4679  -8874  401    -7046 
      {-1} | 3919   -4397  11868  -6952  -5678  -14150 -14747 11774  13423 
      {-1} | -13193 6849   -640   -10622 4101   594    -8593  -13525 -10971
      {-1} | -2982  14611  -14958 -10702 -1647  -9178  -3988  2403   -5779 
      {-1} | -823   5516   -4933  -11174 -11636 2711   -1805  -10796 -13696
      {-1} | -4319  12065  -5230  -8551  -10383 4736   14358  5762   3318  
      {-1} | 6545   -8920  -6035  6000   -14441 -14342 -13953 3366   12406 
      {-1} | 4571   -4655  -714   -1917  -4145  -15436 5034   1876   -6334 
      {-1} | -7429  4894   -6146  10651  1497   -2657  9969   -3129  13819 
      {-1} | 11980  -15408 -438   -9022  -1076  -9845  149    -3983  -7274 
      {-1} | 0      0      0      0      0      0      0      0      0     
      {-1} | 0      0      -8253  0      0      0      0      0      0     
      {-1} | 0      -8253  0      0      0      0      0      0      0     
      {-1} | -8253  0      0      0      0      0      0      0      0     
      -----------------------------------------------------------------------
      -2862  -641   -3249  -15028 -9227  -15894 -13331 -12524 -9822  -14128
      1729   9173   9277   -14846 1158   -3781  13203  12679  -12410 11907 
      9588   4764   -9478  -2982  2071   14322  -9411  3605   14048  -5888 
      1399   2122   -12437 -2779  2652   11718  -14665 9341   -11925 2926  
      12912  -15464 -3513  -11770 12040  2380   -10361 -13243 13248  12550 
      -14009 -14179 11030  12342  -7837  10691  -12187 3899   12834  15540 
      11998  3271   13309  -11969 -7644  6072   -3798  -6357  1119   3837  
      -11398 -14781 13164  -7154  -9803  10817  525    7272   8900   13810 
      -10232 1168   -6301  8645   4330   7746   13503  4636   14600  -15074
      -9390  2436   11660  -15000 10029  -14731 -776   15795  -3934  8918  
      1848   -5596  386    -11687 -2679  -2420  -15458 -974   -7173  -10154
      -8591  2630   145    -15970 13673  12901  -13136 11150  15470  1347  
      -3639  12687  -760   59     9562   -6172  -12892 8284   15132  5982  
      9493   1253   6353   -13858 -15092 9011   -1870  14070  -12753 -13567
      -13447 15996  -13555 14977  8307   2545   -10736 973    5516   5936  
      12265  -3192  -2823  4590   -15975 -1317  10022  -15382 9897   -15639
      12818  11341  -13901 -4843  11959  10453  -54    3938   12674  10464 
      3593   -3765  -5111  -506   13650  3623   -12598 186    7787   -2846 
      -11160 -13788 4048   -2652  -6657  -4860  -15235 -5122  8639   -15716
      153    1780   5403   3926   -2231  8969   -10934 6054   9851   6587  
      -8253  0      0      0      0      0      0      0      0      0     
      0      8253   0      0      0      0      0      0      0      0     
      0      0      0      0      0      0      0      0      0      0     
      0      0      0      0      0      0      0      0      0      0     
      -----------------------------------------------------------------------
      -10639 8538   11196  -563   -1676  0 0 0 0 0 0        0 0       0      
      -7622  572    -9888  7150   13272  0 0 0 0 0 0        0 0       0      
      -790   3687   -10444 13351  -12079 0 0 0 0 0 0        0 0       0      
      -2533  -3644  4222   15931  -15685 0 0 0 0 0 0        0 0       0      
      15274  -9160  15564  1183   -4959  0 0 0 0 0 0        0 0       0      
      -7582  8919   -14464 -10869 14369  0 0 0 0 0 0        0 0       0      
      2014   -11015 -5521  11175  -11780 0 0 0 0 0 0        0 0       0      
      4770   2852   14858  12918  -7949  0 0 0 0 0 0        0 0       0      
      14445  12178  8566   69     -10560 0 0 0 0 0 0        0 0       0      
      1609   -1455  3583   -1284  -3763  0 0 0 0 0 0        0 0       0      
      966    11902  -5370  -4631  -5298  0 0 0 0 0 0        0 0       0      
      -4409  11800  -2116  -7846  -12213 0 0 0 0 0 0        0 0       0      
      10981  1207   -12239 -11077 -6630  0 0 0 0 0 0        0 0       0      
      1795   -585   -694   43     -3702  0 0 0 0 0 0        0 0       0      
      4123   4230   -947   -56    -7949  0 0 0 0 0 0        0 0       0      
      -1426  3250   7426   4451   -3010  0 0 0 0 0 0        0 0       0      
      5691   10982  10375  -13397 10984  0 0 0 0 0 0        0 0       0      
      -6994  -13196 -7353  10509  10944  0 0 0 0 0 0        0 0       0      
      13930  12521  -6313  14194  3517   0 0 0 0 0 0        0 0       0      
      15948  2798   14233  15020  896    0 0 0 0 0 0        0 0       0      
      0      0      0      0      0      0 0 0 0 0 0        0 0       0      
      0      0      0      0      0      0 0 0 0 0 0        0 0       0      
      0      0      0      0      0      0 0 0 0 0 0        0 0       0      
      0      0      0      0      0      0 0 0 0 0 -8253x_0 0 8253x_3 8253x_2
      -----------------------------------------------------------------------
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 0       0       0       0 0       0 0 0       0       0 0 0 0 0 0 0 0
      0 8253x_3 8253x_2 0       0 0       0 0 8253x_3 0       0 0 0 0 0 0 0 0
      0 0       0       8253x_2 0 8253x_3 0 0 0       8253x_3 0 0 0 0 0 0 0 0
      -----------------------------------------------------------------------
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 -8253x_3 0 0 0 0 0 0 |
      0 0 0 -8253x_3 0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |
      0 0 0 0        0 0 0 0 0 0        0 0 0 0 0 0 |

o24 : Matrix T1 <-- T2

Caveat

A negative result means that a random choice of homomorphism was not an isomorphism; especially when the ground field is small, this may not be definitive. Passing Tries => N may be helpful.

See also

Ways to use isIsomorphic:

  • isIsomorphic(Matrix,Matrix)
  • isIsomorphic(Module,Module)

For the programmer

The object isIsomorphic is a method function with options.

The source of this document is in Isomorphism.m2:528:0.