subcyclopclgp(fH, p, {flag = 0}) (1/3)
This function takes two arguments: an abelian number field F (given by an fH argument) and an
odd prime p > 2 not dividing [F : Q]. It returns information about the p-Sylow subgroup
A = AF of the ideal class group of F. An optional flag allows to compute only part of the
structure to save time.
We write A = A+ ⊕ A− according to the eigenvalues of complex conjugation. The function
returns a 6-component vector v and we shall concentrate on v[2] and v[3]:
v[1] is p
v[2] is [E, [e1, . . . , ek]] with E =
P
i ei and e1 > · · · > ek. Meaning that A+ has order
pE and is isomorphic to Z/pe1 × · · · × Z/pek (elementary divisors).
v[3] similarly describes A−.
v[4] gives the structure of Gal(F/Q) (elementary divisors)
v[5] is the number of cyclic subfields K 6= Q contained in F
Atelier 2022 (13/01/2022) – p. 5/13