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 $a_1,a_3\in\{0,1\}$, $a_2\in\{-1,0,1\}$.

**Authors:**

**Knowl status:**

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

**Referred to by:**

- ec.discriminant
- ec.q
- ec.q.discriminant
- ec.q.kodaira_symbol
- ec.q.lmfdb_label
- ec.q.real_period
- ec.q.reduction_type
- ec.simple_equation
- lmfdb/elliptic_curves/elliptic_curve.py (line 373)
- lmfdb/elliptic_curves/templates/ec-curve.html (lines 68-69)
- lmfdb/elliptic_curves/templates/ec-isoclass.html (line 21)

**History:**(expand/hide all)

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