Easy groups (1/4)
For this set of groups, the function can actually compute fields by increasing values of
∆K ∈ [A, B] in a fixed interval. This includes
all transitive permutation groups nTk in degree n 6 5, except S5; the group A5 is only
supported by a table allowing ∆K < 1012 (database of 40,314 fields provided by John
Jones and David Roberts);
all cyclic Cℓ, where ℓ is prime or ℓ ∈ {4, 6, 9};
all dihedral Dℓ, where ℓ prime or ℓ = 4.
For n > 3, all results depend on the truth of the GRH, because of our use of class field theory
and ultimately bnfinit.
Except for A5 and A5(6), fields are computed on the fly. Up to subexponential factors in log B,
the complexity is linear in the output size, i.e. O(B1/a(G)+ε) by Malle’s conjecture, which is a
theorem for many of those fields, for instance Abelian fields or Sn for n 6 5. But it’s not a
theorem for A4 or Dℓ for ℓ > 5 for instance, and the complexity is conjectural in these cases.
PARI/GP day 2021 (02/06/2021) – p. 7/14