Defining polynomial
|
$( x^{5} + 4 x + 11 )^{2} + \left(-8 x - 22\right) ( x^{5} + 4 x + 11 ) + 114260 x^{2} + 88 x - 4084102$
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Invariants
| Base field: | $\Q_{13}$ |
| Degree $d$: | $10$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $5$ |
| Discriminant root field: | $\Q_{13}(\sqrt{13\cdot 2})$ |
| Root number: | $-1$ |
| $\card{ \Gal(K/\Q_{ 13 }) }$: | $10$ |
| This field is Galois and abelian over $\Q_{13}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{13}(\sqrt{13\cdot 2})$, 13.5.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 13.5.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of
\( x^{5} + 4 x + 11 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 13 t \)
$\ \in\Q_{13}(t)[x]$
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Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_{10}$ (as 10T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $5$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | $x^{10} - x^{9} + 34 x^{8} - 34 x^{7} + 430 x^{6} - 430 x^{5} + 2509 x^{4} - 2509 x^{3} + 6964 x^{2} - 6964 x + 9637$ |