Defining polynomial
\(x^{12} + 492 x^{11} + 100878 x^{10} + 11034740 x^{9} + 679393095 x^{8} + 22343681112 x^{7} + 308081214422 x^{6} + 67031087124 x^{5} + 6123506385 x^{4} + 1278059380 x^{3} + 60258731628 x^{2} + 1975336395024 x + 26984212327944\) |
Invariants
Base field: | $\Q_{89}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $6$ |
Residue field degree $f$: | $2$ |
Discriminant exponent $c$: | $10$ |
Discriminant root field: | $\Q_{89}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 89 }) }$: | $12$ |
This field is Galois over $\Q_{89}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{89}(\sqrt{3})$, $\Q_{89}(\sqrt{89})$, $\Q_{89}(\sqrt{89\cdot 3})$, 89.3.2.1 x3, 89.4.2.1, 89.6.4.1, 89.6.5.1 x3, 89.6.5.2 x3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | $\Q_{89}(\sqrt{3})$ $\cong \Q_{89}(t)$ where $t$ is a root of \( x^{2} + 82 x + 3 \) |
Relative Eisenstein polynomial: | \( x^{6} + 89 \) $\ \in\Q_{89}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{5} + 6z^{4} + 15z^{3} + 20z^{2} + 15z + 6$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $D_6$ (as 12T3) |
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: | Not computed |