Partitions of Graphs and Selmer Groups of Elliptic Curves of Neumann-Setzer Type

We consider the elliptic curves E^u : y² = x³ + ux² − 16x and their quadratic twists E^u_n 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 E^u_n 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 E^u_n has rank zero for a positive proportion of squarefree integers n with a fixed number of prime divisors.