An **elliptic curve $E$ over $\mathbb{Q}$** has a Weierstrass equation of the form
$$E : y^2 = x^3 + ax + b$$ with $a, b \in \mathbb{Z}$ such that its discriminant
$$\Delta := −16(4a^3 + 27b^2 ) \not = 0.$$
Note that such an equation is not unique and $E$ has a unique minimal Weierstrass equation.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Michael Bennett on 2019-04-10 17:59:13

**Referred to by:**

- ec.galois_rep_elladic_image
- ec.q.galois_rep
- ec.q.galois_rep_image
- ec.q.optimal
- ec.q.search_input
- ec.q.weil_height
- intro.tutorial
- lfunction.known_degree2
- lfunction.underlying_object
- mf.gl2.history.varieties
- lmfdb/elliptic_curves/elliptic_curve.py (line 117)
- lmfdb/elliptic_curves/elliptic_curve.py (line 166)

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

- 2019-04-10 17:59:13 by Michael Bennett (Reviewed)
- 2019-04-10 17:58:59 by Michael Bennett
- 2019-04-10 17:55:47 by Michael Bennett
- 2019-04-10 17:53:03 by Michael Bennett
- 2019-04-10 17:52:48 by Michael Bennett
- 2019-04-10 17:51:26 by Michael Bennett
- 2018-06-19 22:20:06 by John Jones (Reviewed)

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