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.

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- Last edited by Andrew Sutherland on 2022-07-14 09:03:53

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