Elliptic Curves
Weierstrass Equations and Invariants
General and short Weierstrass forms, the discriminant and j-invariant, and the transformations relating equivalent models.
Engineering / MathematicsElliptic Curves2 min readKV-MATH-0634
A curve admits many Weierstrass equations. The discriminant and the j-invariant are the quantities that distinguish genuinely different curves from different presentations of the same one.
The general form
Reduction to short form
In characteristic not two or three, completing the square and then the cube reduces the general form to the short one. In characteristics two and three the reduction fails and different normal forms are used.
The invariants
- Discriminant
- Non-zero exactly when the curve is smooth. Changes by a twelfth power under admissible change of variables.
- j-invariant
- Invariant under all admissible changes of variables. Two curves over an algebraically closed field are isomorphic exactly when their j-invariants agree.
- Conductor
- Records the bad primes and their reduction types. Finer than the discriminant, which depends on the model.
Minimal models
Over the rationals a curve has a minimal Weierstrass model, with discriminant as small as possible. Computing it is a normalisation step performed before any serious arithmetic.
Twists
Quadratic twists share a j-invariant but differ over the rationals. They have the same behaviour at most primes but can have very different rank, which makes them a standard tool for constructing curves with prescribed properties.
Source. Henri Cohen, A Course in Computational Algebraic Number Theory, Springer GTM 138 — 7.1.3. Structural reference unverified: the source file was not available during authoring; chapter and section numbers are taken from the published edition and have not been checked against a physical copy.