An elliptic curve over $\Q$ is **optimal** if it is an optimal quotient of the corresponding modular curve. Every isogeny class contains a unique optimal curve. For more information, see William Stein's page on optimal quotients.

Optimal curves have a Cremona label whose last component is the number 1, with the exception of class 990h where the optimal curve is 990h3 (number 3). This is a historical accident and has no mathematical significance.

NB It has not yet been proved in all cases that the first curve in each class is optimal; however this is true for all isogeny classes of conductor ${}\le400000$, and for many others (for example whenever the isogeny class consists of only one curve). The current optimality status of each curve is shown on its home page.

**Knowl status:**

- Review status: reviewed
- Last edited by Andrew Sutherland on 2020-10-22 08:15:37

**Referred to by:**

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

- 2020-10-22 08:15:37 by Andrew Sutherland (Reviewed)
- 2020-03-16 13:05:19 by John Cremona
- 2019-09-05 07:29:19 by John Cremona
- 2018-06-19 18:43:53 by John Jones (Reviewed)

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