Defining polynomial
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$( x^{3} + 6 x^{2} + 4 )^{7} + \left(42 x^{2} + 42\right) ( x^{3} + 6 x^{2} + 4 ) + 7$
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Invariants
| Base field: | $\Q_{7}$ |
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| Degree $d$: | $21$ |
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| Ramification index $e$: | $7$ |
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| Residue field degree $f$: | $3$ |
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| Discriminant exponent $c$: | $21$ |
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| Discriminant root field: | $\Q_{7}(\sqrt{7\cdot 3})$ | |
| Root number: | $i$ | |
| $\Aut(K/\Q_{7})$: | $C_1$ | |
| Visible Artin slopes: | $[\frac{7}{6}]$ | |
| Visible Swan slopes: | $[\frac{1}{6}]$ | |
| Means: | $\langle\frac{1}{7}\rangle$ | |
| Rams: | $(\frac{1}{6})$ | |
| Jump set: | undefined | |
| Roots of unity: | $342 = (7^{ 3 } - 1)$ |
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Intermediate fields
| 7.3.1.0a1.1 |
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: |
\( x^{7} + \left(42 t^{2} + 7 t + 7\right) x + 7 \)
$\ \in\Q_{7}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (4 t^2 + 3)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | not computed |
| Galois group: | not computed |
| Inertia group: | not computed |
| Wild inertia group: | not computed |
| Galois unramified degree: | not computed |
| Galois tame degree: | not computed |
| Galois Artin slopes: | not computed |
| Galois Swan slopes: | not computed |
| Galois mean slope: | not computed |
| Galois splitting model: | not computed |