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A projective curve defined over a field F of genus one and with a point rational over F.
An elliptic curve can always be written as a non-singular projective plane cubic curve and this is an equivalent definition.
In general, we can write such a curve in Weierstrass form Y2 = X3 + aX + b, with the cubic in X having distinct roots (that is, non-zero discriminant Delta = -4a3 - 27b).
Over fields of characteristic 2 or 3 we need the more general form Y2 + a XY + a3 Y = X + a2 X2 + a4 X + a6.
In addition to these affine points, there is a single point at infinity, O = (0:1:0) in projective coordinates. It is a point of inflexion for E: that is, the tangent has triple contact.
If K is a field containing the field of definition F, we define E(K) to be the set of points (always including O) with coordinates in K.
The points E(K) form an abelian group with O as zero.
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Let E be an elliptic curve defined over a number field K. The set E(K) of points on E with coordinates in K forms a finitely generated abelian group. The rank of the elliptic curve is the rank of the torsion-free part.
It is not known whether the rank of an elliptic curve defined over Q is bounded. There is an example with rank at least 24.
By contrast, bounds are known on the torsion: see Mazur's Theorem.
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The torsion (number of points of finite order) on an elliptic curve over Q is bounded.
The possible torsion group structures are the cyclic groups of order 1,2,3,4,5,6,7,8,9,10,12 and the non-cyclic 2x2, 2x4, 2x6 and 2x8.
There are infinitely many examples of each.
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Points of finite order on an elliptic curve. Also called torsion points.
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