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1 [made Community Wiki]

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

http://archive.numdam.org/ARCHIVE/JTNB/JTNB_1990__2_1/JTNB_1990__2_1_119_0/JTNB_1990__2_1_119_0.pdf

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