show · ec.simple_equation all knowls · up · search:

Every elliptic curve over a field $k$ whose characteristic is not 2 or 3 has a simplified equation (or short Weierstrass model) of the form $y^2 = x^3 + Ax + B$. When $k=\mathbb Q$ is the field of rational numbers, one can choose $A$ and $B$ to be integers.

For elliptic curves over $\Q$ this model will necessarily have bad reduction at 2, even when $E$ has good reduction at 2; it may also bad reduction at 3 even when the minimal model of $E$ does not.

Knowl status:
  • Review status: reviewed
  • Last edited by Andrew Sutherland on 2022-07-14 09:03:53
Referred to by:
History: (expand/hide all) Differences (show/hide)