Defining polynomial
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$( x^{2} + 6 x + 3 )^{7} + 14 x ( x^{2} + 6 x + 3 )^{2} + 7$
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $14$ |
| Ramification index $e$: | $7$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $16$ |
| Discriminant root field: | $\Q_{7}$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{7})$: | $C_1$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\frac{4}{3}]$ |
| Visible Swan slopes: | $[\frac{1}{3}]$ |
| Means: | $\langle\frac{2}{7}\rangle$ |
| Rams: | $(\frac{1}{3})$ |
| Jump set: | undefined |
| Roots of unity: | $48 = (7^{ 2 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{3})$ |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{2} + 6 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{7} + 14 t x^{2} + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (5 t + 2)$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $588$ |
| Galois group: | $C_7^2:C_{12}$ (as 14T23) |
| Inertia group: | Intransitive group isomorphic to $C_7^2:C_3$ |
| Wild inertia group: | $C_7^2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3}]$ |
| Galois mean slope: | $1.3197278911564627$ |
| Galois splitting model: |
$x^{14} - 7 x^{11} + 28 x^{10} + 35 x^{9} - 42 x^{8} - 82 x^{7} - 49 x^{6} + 343 x^{5} + 252 x^{4} + 224 x^{3} + 196 x^{2} + 21 x + 59$
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