Defining polynomial
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\(x^{12} + 24 x^{11} + 216 x^{10} + 880 x^{9} + 1605 x^{8} + 2064 x^{7} + 6576 x^{6} + 11904 x^{5} + 8307 x^{4} - 50984 x^{3} - 57096 x^{2} + 58128 x + 76871\)
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Invariants
| Base field: | $\Q_{7}$ |
| Degree $d$: | $12$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $9$ |
| Discriminant root field: | $\Q_{7}(\sqrt{7})$ |
| Root number: | $-i$ |
| $\card{ \Aut(K/\Q_{ 7 }) }$: | $6$ |
| This field is not Galois over $\Q_{7}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{7}(\sqrt{7\cdot 3})$, 7.3.0.1, 7.4.3.1, 7.6.3.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of
\( x^{3} + 6 x^{2} + 4 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^{3} + 4z^{2} + 6z + 4$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times D_4$ (as 12T14) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $6$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: |
$x^{12} + 15 x^{10} - 4 x^{9} + 45 x^{8} - 3 x^{7} - 81 x^{6} + 36 x^{5} + 264 x^{4} + 104 x^{3} + 108 x^{2} + 48 x + 8$
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