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

<p>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&#39;s bijection [2] instead of Davenport-Heilbronn&#39;s.</p> <p>File &quot;raw&quot;: the following data are given for each field:</p> <ul> <li>Coefficients of a pair (F₁, F₂) of ternary quadratic forms corresponding to the field under Bhargava&#39;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.</li> <li>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.</li> <li>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.</li> </ul> <p>Files &quot;Ti.gp&quot;: the index i is the number of pairs of complex embeddings. The following data are given for each field:</p> <ul> <li>Discriminant.</li> <li>Coefficients of the canonical defining polynomial for the field, as given by PARI&#39;s function polredabs.</li> <li>Number of elements and cyclic decomposition of the class group, as given by PARI&#39;s function bnfinit.</li> </ul> <p>The following sanity checks have been performed:</p> <ul> <li>The number of totally real fields agrees with that computed by Malle [3] using Hunter&#39;s method.</li> <li>The list of discriminants (with multiplicities) agrees with that computed by PARI&#39;s function nflist using class field theory (conditionally on the generalised Riemann hypothesis).</li> </ul> <p>References:</p> <ol> <li>Karim Belabas. A fast algorithm to compute cubic fields. <em>Math. Comp.</em>, 66(219):1213&ndash;1237, 1997.</li> <li>Manjul Bhargava. Higher composition laws. III. The parametrization of quartic rings. <em>Ann. of Math. (2)</em>, 159(3):1329&ndash;1360, 2004.</li> <li>Gunter Malle. The totally real primitive number fields of discriminant at most 10⁹. In <em>Algorithmic number theory</em>, volume 4076 of <em>Lecture Notes in Comput. Sci.</em>, pages 114&ndash;123. Springer, Berlin, 2006.</li> </ol>