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If ''p'' is a [[prime number]] and ''E'' is an elliptic curve over '''Q''' (the [[field (mathematics)|field]] of [[rational number]]s), we can reduce the equation defining ''E'' [[modular arithmetic|modulo]] ''p''; for all but finitely many values of ''p'' we will get an elliptic curve over the [[finite field]] '''F'''<sub>''p''</sub>, with ''n''<sub>''p''</sub> elements, say. One then considers the sequence
:''a''<sub>''p''</sub> = ''n''<sub>''p''</sub> &
which is an important invariant of the elliptic curve ''E''. Every modular form also gives rise to a sequence of numbers, by [[Fourier transform]]. An elliptic curve whose sequence agrees with that from a modular form is called '''modular'''. The Taniyama-Shimura theorem states:
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