We construct an elliptic curve over Q with non-trivial 2-torsion point and rank exactly equal to 15. Introduction Let E be an elliptic curve over Q. By Mordell's theorem, E(Q) is a finitely generated abelian group. This means that E(Q) E(Q) tors × Z r . By Mazur's theorem, we know that E(Q) tors is one of the following 15 groups: Z/nZ with 1 ≤ n ≤ 10 or n = 12, On the other hand, we do not know what values of rank r are possible for elliptic curves over Q. The "folklore" conjecture is that a

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... k can be arbitrary large, but at present only an example of elliptic curve with rank ≥ 24 is known [10]. There is even a stronger conjecture that, for any of 15 possible torsion groups T , we have B(T ) = ∞, where B(T ) = sup{rank E(Q) : torsion group of E over Q is T }. It follows from results of Montgomery [13] and Atkin-Morain [1] that B(T ) ≥ 1 for all torsion groups T . Womack [16] proved that B(T ) ≥ 2 for all T , while Dujella [5] proved that B(T ) ≥ 3 for all T . The best known lower bounds for B(T ) can be found at [5] . In the present paper, we describe the construction of an elliptic curve over Q with a nontrivial 2-torsion point whose rank is exactly equal to 15. This gives the current record for the rank of curves with nontrivial torsion, and also the highest example of the rank of an elliptic curve which is known exactly (not only a lower bound for rank). It improves previous records due 0 2000 Mathematics Subject Classification: 11G05.