Defining polynomial
|
\(x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504\)
|
Invariants
| Base field: | $\Q_{2}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $14$ |
| Discriminant root field: | $\Q_{2}(\sqrt{-1})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 2 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{2}.$ | |
| Visible slopes: | $[2]$ |
Intermediate fields
| $\Q_{2}(\sqrt{-1})$, 2.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of
\( x^{7} + x + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 2 x + 4 t^{5} + 4 t^{4} + 4 t^{3} + 4 t^{2} + 4 t + 2 \)
$\ \in\Q_{2}(t)[x]$
|
Ramification polygon
| Residual polynomials: | $z + 1$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_2$ |
| Unramified degree: | $7$ |
| Tame degree: | $1$ |
| Wild slopes: | $[2]$ |
| Galois mean slope: | $1$ |
| Galois splitting model: | $x^{14} + 42 x^{12} + 623 x^{10} + 4431 x^{8} + 16513 x^{6} + 31906 x^{4} + 28784 x^{2} + 9409$ |