show · ec.q.optimal all knowls · up · search:

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) Differences (show/hide)