Defining polynomial
\(x^{12} - 1586 x^{6} - 198575\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{13}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{2})$, 13.3.2.2, 13.4.2.2, 13.6.4.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} + 12 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{6} + 156 t + 143 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + 2z^{3} + 7z^{2} + 2z + 6$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{12}$ (as 12T1) |
Inertia group: | Intransitive group isomorphic to $C_6$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $6$ |
Wild slopes: | None |
Galois mean slope: | $5/6$ |
Galois splitting model: | $x^{12} - x^{11} + 10 x^{10} - 15 x^{9} + x^{8} - 55 x^{7} + 24 x^{6} + 155 x^{5} + 471 x^{4} + 505 x^{3} + 415 x^{2} + 309 x + 521$ |