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:

1. Karim Belabas. A fast algorithm to compute cubic fields. Math. Comp., 66(219):1213–1237, 1997.
2. Manjul Bhargava. Higher composition laws. III. The parametrization of quartic rings. Ann. of Math. (2), 159(3):1329–1360, 2004.
3. 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.