Defining polynomial
\(x^{12} + 15 x^{10} + 40 x^{9} + 84 x^{8} + 120 x^{7} + 53 x^{6} + 414 x^{5} - 1293 x^{4} - 1830 x^{3} + 10968 x^{2} - 13836 x + 12004\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{7}(\sqrt{3})$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{3})$, 7.3.2.2, 7.4.0.1, 7.6.4.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 7.4.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{4} + 5 x^{2} + 4 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{3} + 7 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_{12}$ (as 12T1) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $4$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{12} - x^{11} + 3 x^{10} - 4 x^{9} + 9 x^{8} + 2 x^{7} + 12 x^{6} + x^{5} + 25 x^{4} - 11 x^{3} + 5 x^{2} - 2 x + 1$ |