Many class group computations are sped up tremendously by assuming the GRH. As I understand it this is done by computing upper bounds on the discriminants of potential abelian extensions. See this survey by Odlyzko for more details
This is built into SAGE.
sage: J=JonesDatabase() sage: NFs=J.unramified_outside([2,3]) sage: time RHCNs = [K.class_number(proof=False) for K in NFs] CPU times: user 7.05 s, sys: 0.07 s, total: 7.13 s Wall time: 7.15 s sage: time CNs = [K.class_number() for K in NFs] CPU times: user 20.19 s, sys: 0.24 s, total: 20.43 s Wall time: 20.96 s