Defining polynomial
\(x^{12} + 140 x^{10} + 18 x^{9} + 8092 x^{8} + 20 x^{7} + 244001 x^{6} - 56140 x^{5} + 4059563 x^{4} - 1721304 x^{3} + 36312468 x^{2} - 14694000 x + 141161628\) |
Invariants
Base field: | $\Q_{23}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{23}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 23 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{23}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{23}(\sqrt{5})$, $\Q_{23}(\sqrt{23})$, $\Q_{23}(\sqrt{23\cdot 5})$, 23.3.0.1, 23.4.2.1, 23.6.0.1, 23.6.3.1, 23.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 23.6.0.1 $\cong \Q_{23}(t)$ where $t$ is a root of \( x^{6} + x^{4} + 9 x^{3} + 9 x^{2} + x + 5 \) |
Relative Eisenstein polynomial: | \( x^{2} + 23 \) $\ \in\Q_{23}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times C_6$ (as 12T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | Not computed |