Defining polynomial
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$( x^{2} + 12 x + 2 )^{4} + \left(-48 x + 856\right) ( x^{2} + 12 x + 2 )^{3} + \left(-16992 x + 98650\right) ( x^{2} + 12 x + 2 )^{2} + \left(-1427568 x + 2599800\right) ( x^{2} + 12 x + 2 ) - 33132960 x - 5602585$
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Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $8$ |
| Ramification exponent $e$: | $4$ |
| Residue field degree $f$: | $2$ |
| Discriminant exponent $c$: | $6$ |
| Discriminant root field: | $\Q_{13}$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $8$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{2})$, $\Q_{13}(\sqrt{13})$, $\Q_{13}(\sqrt{13\cdot 2})$, 13.4.2.1, 13.4.3.4, 13.4.3.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{2} + 12 x + 2 \)
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| Relative Eisenstein polynomial: |
\( x^{4} + 13 t + 143 \)
$\ \in\Q_{13}(t)[x]$
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Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_2\times C_4$ (as 8T2) |
| Inertia group: | Intransitive group isomorphic to $C_4$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $2$ |
| Tame degree: | $4$ |
| Wild slopes: | None |
| Galois mean slope: | $3/4$ |
| Galois splitting model: | $x^{8} - x^{7} - 21 x^{6} + 18 x^{5} + 89 x^{4} - 19 x^{3} - 89 x^{2} - 38 x - 4$ |