Defining polynomial
\(x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009\) |
Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $2$ |
Residue field degree $f$: | $6$ |
Discriminant exponent $c$: | $6$ |
Discriminant root field: | $\Q_{5}$ |
Root number: | $-1$ |
$\card{ \Gal(K/\Q_{ 5 }) }$: | $12$ |
This field is Galois and abelian over $\Q_{5}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{5}(\sqrt{2})$, $\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, 5.3.0.1, 5.4.2.1, 5.6.0.1, 5.6.3.1, 5.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 5.6.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{6} + x^{4} + 4 x^{3} + x^{2} + 2 \) |
Relative Eisenstein polynomial: | \( x^{2} + 20 x + 5 \) $\ \in\Q_{5}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z + 2$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_2\times C_6$ (as 12T2) |
Inertia group: | Intransitive group isomorphic to $C_2$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $6$ |
Tame degree: | $2$ |
Wild slopes: | None |
Galois mean slope: | $1/2$ |
Galois splitting model: | $x^{12} - x^{11} + 2 x^{10} - 3 x^{9} + 5 x^{8} - 8 x^{7} + 13 x^{6} + 8 x^{5} + 5 x^{4} + 3 x^{3} + 2 x^{2} + x + 1$ |