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The LMFDB label of an elliptic curve $E$ over $\mathbb{Q}$ is a way of indexing the elliptic curves over $\mathbb Q.$ It has the form "11.a1" or "10050.bf2".

The label has three components: the conductor, the isogeny class label, and the isomorphism class index.

  1. The first component is the decimal representation of the conductor (a positive integer).

  2. The second component is the isogeny class label, a string which represents the isogeny class index, a non-negative integer encoded as in base 26 using the 26 symbols a,b,.., z. The isogeny classes of elliptic curves with the same conductor are sorted lexicographically by the $q$-expansions of the associated modular forms, and the isogeny class index of each isogeny class of fixed conductor is the index (starting at 0) of the class in this ordering.

  3. The third component is the decimal representation of the isomorphism class index, a positive integer giving the index of the coefficient vector $[a_1, a_2, a_3, a_4, a_6]$ of the reduced minimal Weierstrass equation of $E$ in a lexicographically sorted list of all the elliptic curves in the isogeny class.

The complete label is obtained by concatenating [conductor, ".", isogeny class label, isomorphism class index].

Note that this is not the same as the Cremona label, even though for certain curves they only differ in the insertion of the dot "." (for example, "37a1" and "37.a1" are the same curve). The presence of the punctuation "." distinguishes an LMFDB label from a Cremona label. Cremona labels are only defined for curves of conductor up to 500000.