negative -- implements the inverse in the group law of a curve of genus 1

• Usage:

  q = negative(o,p,E)

  qE = negative(oE,pE)

• Inputs:
  • o, a list,
  • oE, an ideal,
  • p, a list,
  • pE, an ideal,
  • E, a ring,
• Optional inputs:
  • Conductor => ..., default value null
• Outputs:
  • q, a list,
  • qE, an ideal,

i1 : S = QQ[x,y,z]

o1 = S

o1 : PolynomialRing

i2 : E = S/ideal"x3+y3+z3"

o2 = E

o2 : QuotientRing

i3 : o = {1,-1,0}

o3 = {1, -1, 0}

o3 : List

i4 : p = {0,1,-1}

o4 = {0, 1, -1}

o4 : List

i5 : negative(o,p,E)

o5 = {-1, 0, 1}

o5 : List

i6 : q = addition(o,p,p,E)

o6 = {-1, 0, 1}

o6 : List

i7 : r = addition(o,q, negative(o,p,E), E)

o7 = {0, -1, 1}

o7 : List

i8 : r == p

o8 = false