Commit 2024-10-04 20:24 811c9e37
View on Github →feat(AlgebraicGeometry/EllipticCurve/NormalForms): some normal forms of elliptic curves (#16864) This PR adds some normal forms of elliptic curves by characteristic. The results are from Silverman's GTM106 second edition, section III.1, page 42, and Appendix A, Proposition 1.1.
- Normal form of characteristics not 2:
Y^2 = X^3 + a_2X^2 + a_4X + a_6. - Normal form of characteristics not 2 or 3:
Y^2 = X^3 + a_4X + a_6. This is also the normal form of characteristic = 3 and j = 0. - Normal form of characteristic = 3 and j not 0:
Y^2 = X^3 + a_2X^2 + a_6. - Normal form of characteristic = 2 and j = 0:
Y^2 + a_3Y = X^3 + a_4X + a_6. - Normal form of characteristic = 2 and j not 0:
Y^2 + XY = X^3 + a_2X^2 + a_6. Basic properties and the existence of the change of variables to change a curve into these forms are included. Legendre normal form and Deuring normal form may be added in this file in the future PRs.