A **Weierstrass equation** or **Weierstrass model** over a field $k$ is a plane curve $E$ of the form
$$y^2 + a_1xy + a_3y = x^3 + a_2 x^2 + a_4 x + a_6,$$
with $a_1, a_2, a_3, a_4, a_6 \in k$.

The **Weierstrass coefficients** of this model $E$ are the five coefficients $a_i$. These are often displayed as a list $[a_1, a_2, a_3, a_4, a_6]$.

It is common not to distinguish between the *affine* curve defined by a Weierstrass equation and its *projective closure*, which contains exactly one additional *point at infinity*, $[0:1:0]$.

A Weierstrass model is smooth if and only if its discriminant $\Delta$ is nonzero. In this case, the plane curve $E$ together with the point at infinity as base point, define an elliptic curve defined over $k$.

Two smooth Weierstrass models define isomorphic elliptic curves if and only if they are isomorphic as Weierstrass models.

**Knowl status:**

- Review status: reviewed
- Last edited by Michael Bennett on 2019-04-10 17:47:42

**Referred to by:**

- ec
- ec.congruent_number_curve
- ec.discriminant
- ec.global_minimal_model
- ec.integral_model
- ec.invariants
- ec.isomorphism
- ec.j_invariant
- ec.local_minimal_model
- ec.period
- ec.q.j_invariant
- ec.q.period_lattice
- ec.q.real_period
- ec.q.search_input
- ec.weierstrass_isomorphism
- lmfdb/ecnf/main.py (lines 396-401)
- lmfdb/ecnf/templates/ecnf-curve.html (line 52)
- lmfdb/ecnf/templates/ecnf-isoclass.html (line 58)
- lmfdb/elliptic_curves/elliptic_curve.py (line 482)
- lmfdb/modular_curves/web_curve.py (line 201)
- lmfdb/modular_curves/web_curve.py (line 629)

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

- 2019-04-10 17:47:42 by Michael Bennett (Reviewed)
- 2018-06-18 00:25:38 by John Jones (Reviewed)

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