Every elliptic curve over $\mathbb{Q}$ has an integral Weierstrass model (or equation) of the form \[y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6,\]
where $a_1,a_2,a_3,a_4,a_6$ are integers.
Each such equation has a discriminant $\Delta$. A **minimal Weierstrass equation** is one for which $|\Delta|$ is minimal among all Weierstrass models for the same curve. For elliptic curves over $\mathbb{Q}$, minimal models exist, and there is a unique **reduced minimal model** which satisfies the additional constraints $a_1,a_3\in\{0,1\}$, $a_2\in\{-1,0,1\}$.

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- Last edited by John Jones on 2018-06-18 21:15:42

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- 2018-06-18 21:15:42 by John Jones (Reviewed)