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Authors
Affiliations
1 University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, PL
Source
Journal of the Ramanujan Mathematical Society, Vol 29, No 1 (2014), Pagination: 75-92
Abstract
We consider the elliptic curves Eu : y2 = x3 + ux2 − 16x and their quadratic twists Eun by a squarefree integer n, where u2 + 64 = p1 . . . pl, (pi are primes). When l ≤ 2, n ≡ 1(mod 4) and all prime divisors of n are congruent to 3 modulo 4 we give a complete description of sizes of Selmer groups of Eun in terms of number of even partitions of some graphs. If n is even or l > 2, we give some conditions for twists of rank zero. We deduce also that Eun has rank zero for a positive proportion of squarefree integers n with a fixed number of prime divisors.