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The [[theorem]] states that any [[elliptic curve]] over [[Rational number|{{<math|ℚ}}]]>\Q</math> can be obtained via a [[rational map]] with [[integer]] [[coefficient]]s from the [[classical modular curve]] {{math|''X''<sub>0</sub>(''N'')}} for some integer {{mvar|N}}; this is a curve with integer coefficients with an explicit definition. This mapping is called a modular parametrization of level {{mvar|N}}. If {{mvar|N}} is the smallest integer for which such a parametrization can be found (which by the modularity theorem itself is now known to be a number called the ''[[conductor of an elliptic curve|conductor]]''), then the parametrization may be defined in terms of a mapping generated by a particular kind of modular form of weight two and level {{mvar|N}}, a normalized [[newform]] with integer {{mvar|q}}-expansion, followed if need be by an [[Elliptic curve#Isogeny|isogeny]].
===Related statements===
The modularity theorem implies a closely related analytic statement:
To each elliptic curve {{mvar|E}} over {{<math|ℚ}}>\Q</math> we may attach a corresponding [[L-series of an elliptic curve|{{mvar|L}}-series]]. The {{mvar|L}}-series is a [[Dirichlet series]], commonly written
