Abstract. We construct a family of Diophantine triples {c 1 (t), c 2 (t), c 3 (t)} such that the elliptic curve over Q(t) induced by this triple, i.e.:has torsion group isomorphic to Z/2Z × Z/2Z and rank 5. This represents an improvement of the result of A. Dujella, who showed a family of this kind with rank 4. By specialization we obtain two examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and rank equal to 11. This is also an improvement over the known results relating this kind of curves.
We construct an elliptic curve over the field of rational functions with torsion group Z/2Z×Z/4Z and rank equal to four, and an elliptic curve over Q with the same torsion group and rank nine. Both results improve previous records for ranks of curves of this torsion group. They are obtained by considering elliptic curves induced by Diophantine triples.
Abstract. Triangles having rational sides a, b, c and rational area Q are called Heron triangles. Associated to each Heron triangle is the quarticThe Heron formula states that Q = √ P (P − a)(P − b) (P − c) where P is the semi-perimeter of the triangle, so the point (u, v) = (P, Q) is a rational point on the quartic. Also the point of innity is on the quartic. By a standard construction it can be proved that the quartic is equivalent to the elliptic curveThe point (P, Q) on the quartic transforms toon the cubic, and the point of innity goes to (0, abc). Both points are independent, so the family of curves induced by Heron triangles has rank ≥ 2. In this note we construct subfamilies of rank at least 3, 4 and 5. For the subfamily with rank ≥ 5, we show that its generic rank is exactly equal to 5 and we nd free generators of the corresponding group. By specialization, we obtain examples of elliptic curves over Q with rank equal to 9 and 10. This is an improvement of results by F. Izadi et al., who found a subfamily with rank ≥ 3, and several examples of curves of rank 7 over Q. The Heron formula states that the area Q of a triangle with sides {a, b, c} is equal toSo, a triangle with sides a, b, c is a Heron triangle when
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