The S₅Extensions of Degree 6 with Minimum Discriminant

A Database for Field Extensions of the Rationals

“…The minimal discriminants for all signatures of degree 6 are −184607 612233 −2306599 20134393 computed in [30]. Olivier [26] and Ford and Pohst [17,18,19] have completed the computation of the minimal discriminants of all signatures and all primitive Galois groups of degree 6. Olivier and others [27,4] have also computed the minimal fields for imprimitive groups of degree 6.…”

The tame-wild principle for discriminant relations for number fields

“…From (2.4), one sees that to find all K 6 /F with |D K6/F | ≤ B it suffices to find all K 5 /F with |D K5/F | ≤ B, apply the Cayley-Weber resolvent, and keep those K 6 /F with |D K6/F | ≤ B. This indirect quintic method is enormously faster for large B, but the direct sextic method over F = Q was used in [FP92] and [FPDH98] for the P SL 2 (5) and P GL 2 (5) cases respectively.…”

Mixed degree number field computations

“…Our second illustration of the method of mixed degree searches comes from the sextic groups PSL 2 (5) = 6T 12 and PGL 2 (5) = 6T 14. The fields with smallest root discriminant were obtained by a direct sextic search in [FP92] and [FPDH98]. These root discriminants are 2 1 67 1/3 ≈ 8.12 and 2 1 3 1/3 7 1/2 ≈ 11.01.…”