Primitive quartic number fields of absolute discriminant at most 10^9

Complete list of all primitive number fields of degree 4 and absolute discriminant at most 109. Computed through a method similar to that of Belabas [1], but starting from Bhargava's bijection [2] instead of Davenport-Heilbronn's.

File "raw": the following data are given for each field:

Coefficients of a pair (F1, F2) of ternary quadratic forms corresponding to the field under Bhargava's bijection: F1 = v0 x2 + v1 x y + v2 x z + v3 y2 + v4 y z + v5 z2 and F2 = v6 x2 + v7 x y + v8 x z + v9 y2 + v10 y z + v11 z2.

Cubic covariant of (F1, F2): if M1, M2 are the matrices representing F1, F2, then 4 det(x M1 + y M2) = p0 x3 + p1 x2 y + p2 x y2 + p3 y3.

Resultant of F1(x, y, 1) and F2(x, y, 1) with respect to y: r0 x4 + r1 x3 + r2 x2 + r3 x + r4. This is a defining polynomial for the field.

Files "Ti.gp": the index i is the number of pairs of complex embeddings. The following data are given for each field:

Discriminant.

Coefficients of the canonical defining polynomial for the field, as given by PARI's function polredabs.

Number of elements and cyclic decomposition of the class group, as given by PARI's function bnfinit.

The following sanity checks have been performed:

The number of totally real fields agrees with that computed by Malle [3] using Hunter's method.

The list of discriminants (with multiplicities) agrees with that computed by PARI's function nflist using class field theory (conditionally on the generalised Riemann hypothesis).

References:

Karim Belabas. A fast algorithm to compute cubic fields. Math. Comp., 66(219):1213–1237, 1997.

Manjul Bhargava. Higher composition laws. III. The parametrization of quartic rings. Ann. of Math. (2), 159(3):1329–1360, 2004.

Gunter Malle. The totally real primitive number fields of discriminant at most 109. In Algorithmic number theory, volume 4076 of Lecture Notes in Comput. Sci., pages 114–123. Springer, Berlin, 2006.