Defining polynomial
\(x^{14} + 910 x^{13} + 354991 x^{12} + 76965980 x^{11} + 10019422049 x^{10} + 783716146850 x^{9} + 34178120507901 x^{8} + 647835779318542 x^{7} + 444315567676253 x^{6} + 132448262974664 x^{5} + 22043433979203 x^{4} + 4640995086910 x^{3} + 109804606053970 x^{2} + 2252387433208628 x + 7052605388282809\) |
Invariants
Base field: | $\Q_{13}$ |
Degree $d$: | $14$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $7$ |
Discriminant exponent $c$: | $7$ |
Discriminant root field: | $\Q_{13}(\sqrt{13})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 13 }) }$: | $14$ |
This field is Galois and abelian over $\Q_{13}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{13}(\sqrt{13})$, 13.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 13.7.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{7} + 3 x + 11 \) |
Relative Eisenstein polynomial: | \( x^{2} + 130 x + 13 \) $\ \in\Q_{13}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_{14}$ (as 14T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $7$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |