EllipticCurves -- elliptic curves and a brute-force method for point counting

Authors

Alessandro Oneto <[email protected]>
Stefano Marseglia <[email protected]>

Version

This documentation describes version 0.1 of EllipticCurves.

Source code

The source code from which this documentation is derived is in the file EllipticCurves.m2.

Exports

Types
- EllipticCurve -- The class of elliptic curves in short Weierstrass form.
- EllipticCurveW -- The class of elliptic curves in Weierstrass form.
- EllPoint -- The class of points of elliptic curves in short Weierstrass form.
- EllPointW -- The class of points of elliptic curves in Weierstrass form.
Functions and commands
- bruteForceGroupOrder -- A brute-force method to count the points on an elliptic curve in characteristic different from 2.
- bruteForceRationalPoints -- A brute-force method to list the points on an elliptic curve.
- Discriminant -- A method to compute the discriminant of an elliptic curve.
- ellCurve -- A method for creating elliptic curves.
- ellCurveFromjInv -- A method to define an elliptic curve from a given j-invariant.
- ellPoint -- A method for defining point on an elliptic curve.
- isElliptic -- A method to check if an ideal defines an elliptic curve.
- isOnEllCurve -- A method to check if a point is on an elliptic curve.
- jInvariant -- A method to compute the j-invariant of an elliptic curve.
- toShortWForm -- A method to transform an elliptic curve from Weierstrass form to short Weierstrass form.
- toWeierstrass -- A method to transform a smooth plane cubic into Weierstrass form.
Methods
- - EllPoint
- "- EllPointW" -- see - EllPoint
- bruteForceGroupOrder(EllipticCurve) -- A brute-force method to count the points on an elliptic curve in characteristic different from 2.
- bruteForceGroupOrder(EllipticCurveW) -- A brute-force method to count the points on an elliptic curve in characteristic different from 2.
- bruteForceRationalPoints(EllipticCurve) -- A brute-force method to list the points on an elliptic curve.
- bruteForceRationalPoints(EllipticCurveW) -- A brute-force method to list the points on an elliptic curve.
- Discriminant(EllipticCurve) -- A method to compute the discriminant of an elliptic curve.
- Discriminant(EllipticCurveW) -- A method to compute the discriminant of an elliptic curve.
- ellCurve(List,Thing) -- A method for creating elliptic curves in Weierstrass form.
- ellCurve(Thing,Thing,Thing) -- A method for creating elliptic curves in short Weierstrass form.
- "ellCurveFromjInv(Thing)" -- see ellCurveFromjInv -- A method to define an elliptic curve from a given j-invariant.
- EllipticCurve == EllipticCurve
- "EllipticCurveW == EllipticCurveW" -- see EllipticCurve == EllipticCurve
- EllPoint + EllPoint
- "EllPointW + EllPointW" -- see EllPoint + EllPoint
- EllPoint - EllPoint
- "EllPointW - EllPointW" -- see EllPoint - EllPoint
- EllPoint == EllPoint
- "EllPointW == EllPointW" -- see EllPoint == EllPoint
- ellPoint(List,EllipticCurve) -- A method for defining point on an elliptic curve.
- ellPoint(List,EllipticCurveW) -- A method for defining point on an elliptic curve.
- "isElliptic(Ideal)" -- see isElliptic -- A method to check if an ideal defines an elliptic curve.
- isOnEllCurve(List,EllipticCurve) -- A method to check if a point is on an elliptic curve.
- isOnEllCurve(List,EllipticCurveW) -- A method to check if a point is on an elliptic curve.
- jInvariant(EllipticCurve) -- A method to compute the j-invariant of an elliptic curve.
- jInvariant(EllipticCurveW) -- A method to compute the j-invariant of an elliptic curve.
- "expression(EllipticCurve)" -- see net(EllipticCurve)
- "expression(EllipticCurveW)" -- see net(EllipticCurve)
- net(EllipticCurve)
- "net(EllipticCurveW)" -- see net(EllipticCurve)
- "expression(EllPoint)" -- see net(EllPoint)
- "expression(EllPointW)" -- see net(EllPoint)
- net(EllPoint)
- "net(EllPointW)" -- see net(EllPoint)
- "toShortWForm(EllipticCurveW)" -- see toShortWForm -- A method to transform an elliptic curve from Weierstrass form to short Weierstrass form.
- "toWeierstrass(List,List,Thing)" -- see toWeierstrass -- A method to transform a smooth plane cubic into Weierstrass form.
- "EllPoint * ZZ" -- see ZZ * EllPoint
- "EllPointW * ZZ" -- see ZZ * EllPoint
- ZZ * EllPoint
- "ZZ * EllPointW" -- see ZZ * EllPoint

For the programmer

The object EllipticCurves is a package.