Defining polynomial
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$( x^{3} + 6 x^{2} + 4 )^{4} + 7 x$
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification index $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{7})$: | $C_6$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible Artin slopes: | $[\ ]$ |
| Visible Swan slopes: | $[\ ]$ |
| Means: | $\langle\ \rangle$ |
| Rams: | $(\ )$ |
| Jump set: | undefined |
| Roots of unity: | $342 = (7^{ 3 } - 1)$ |
Intermediate fields
| $\Q_{7}(\sqrt{7})$, 7.3.1.0a1.1, 7.1.4.3a1.2, 7.3.2.3a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 7.3.1.0a1.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{3} + 6 x^{2} + 4 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + 7 t \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^3 + 4 z^2 + 6 z + 4$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois degree: | $24$ |
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\ ]$ |
| Galois Swan slopes: | $[\ ]$ |
| Galois mean slope: | $0.75$ |
| Galois splitting model: |
$x^{12} - 6 x^{11} + 15 x^{10} - 16 x^{9} + 12 x^{8} - 66 x^{7} - 67 x^{6} + 942 x^{5} - 3261 x^{4} + 4534 x^{3} - 408 x^{2} - 1692 x - 159$
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