Defining polynomial
\(x^{12} - 154 x^{6} - 1421\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
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, 7.4.2.2, 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: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{6} + 35 t + 28 \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_{12}$ (as 12T1) |
Inertia group: | Intransitive group isomorphic to $C_6$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $6$ |
Wild slopes: | None |
Galois mean slope: | $5/6$ |
Galois splitting model: | $x^{12} - x^{11} - 12 x^{10} + 11 x^{9} + 54 x^{8} - 43 x^{7} - 113 x^{6} + 71 x^{5} + 110 x^{4} - 46 x^{3} - 40 x^{2} + 8 x + 1$ |