Defining polynomial
\(x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372\) |
Invariants
Base field: | $\Q_{7}$ |
Degree $d$: | $6$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $3$ |
Discriminant exponent $c$: | $3$ |
Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
Root number: | $i$ |
$\card{ \Gal(K/\Q_{ 7 }) }$: | $6$ |
This field is Galois and abelian over $\Q_{7}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{7}(\sqrt{7})$, 7.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{3} + 6 x^{2} + 4 \) |
Relative Eisenstein polynomial: | \( x^{2} + \left(28 t^{2} + 42 t + 35\right) x + 7 t \) $\ \in\Q_{7}(t)[x]$ |
Ramification polygon
Data not computedInvariants of the Galois closure
Galois group: | $C_6$ (as 6T1) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $3$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{6} - x^{5} + 21 x^{4} - 22 x^{3} + 58 x^{2} + 23 x + 155$ |