GRH and the rank of elliptic curves I have been using the Magma calculator recently, and while calculating ranks of elliptic curves with very big coefficients, there is a possibility to assume GRH is true, which signaficantly speeds up the calculation.
My question is, how is computation of the rank of an elliptic curve made faster by assuming the GRH. I am no expert in this field, so please keep your answers as simple as possible.
 A: Computation of ranks of elliptic curves relies on descent. The first step of descent is the computation of a finite Selmer group, which in turn uses the computation of the class group of a potentially large number field. This is the step where GRH is used: it allows you to assume that the class group is generated by the set of prime ideals up to a relatively small norm bound, therefore speeding up the computation.
A: Note: Here i present the method assuming GRH when the rank is large compared to the conductor .may this helping you .
Take $f(x)$ to be a function such that $f(0)=1$ and $f(x)\geq0$ for all real $x$
Then, assuming the Riemann hypothesis, the sum :$\sum f(\beta)$ where $1/2+i\beta$ runs over the nontrivial zeros of $L(s,E)$,will be an upper bound for the analytic rank of $E$ Moreover, for certain choices of $f(x)$  this sum may be efficiently evaluated using the explicit formula for the $L$-function attached to $E$ , The method, is available as part of William Stein’s
   (W. A. Stein et al., Purple PSAGE, The PSAGE Development Team, 2011,)
