Defining polynomial
\(x^{8} + 28 x^{7} + 302 x^{6} + 1540 x^{5} + 3623 x^{4} + 3388 x^{3} + 4178 x^{2} + 13468 x + 22324\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $8$ |
Ramification exponent $e$: | $4$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{11}$ |
Root number: | $1$ |
$\card{ \Gal(K/\Q_{ 11 }) }$: | $8$ |
This field is Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, $\Q_{11}(\sqrt{11})$, $\Q_{11}(\sqrt{11\cdot 2})$, 11.4.2.1, 11.4.3.1 x2, 11.4.3.2 x2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{2} + 7 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{4} + 11 \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_4$ (as 8T4) |
Inertia group: | Intransitive group isomorphic to $C_4$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $2$ |
Tame degree: | $4$ |
Wild slopes: | None |
Galois mean slope: | $3/4$ |
Galois splitting model: | $x^{8} - x^{7} + 2 x^{6} - 3 x^{5} - x^{4} + 6 x^{3} + 8 x^{2} + 8 x + 16$ |