Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Can we go from short Weierstrass equation equation $y^2=x^3+Ax^2+Bx+C$ to general $$y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6$$?

share|cite|improve this question

What you call the "general equation" is the Weierstrass equation, and what you call the "Weierstrass equation" is the so-called "short" Weierstrass equation, which is just the general Weierstrass equation with some coefficients $0$, so your question doesn't really make sense.

But maybe you meant to ask it the other way around: how to get a short Weierstrass equation from a general one? We perform a change of variables which preserves the behavior of the curve, called an admissible change of variables. If we are over a field of characteristic not $2$ or $3$, it is always possible to get a short equation from a general one through an admissible change of variables. Otherwise, not always.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.