Defining polynomial
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$( x^{7} + 2 x^{2} + 1 )^{2} + \left(21 x^{5} + 189 x^{3} + 945 x - 138\right) ( x^{7} + 2 x^{2} + 1 ) + 2835 x^{6} - 1029 x^{5} + 5103 x^{4} - 1449 x^{3} + 5403 x^{2} + 297 x + 2349$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $14$ |
| Ramification exponent $e$: | $2$ |
| Residue field degree $f$: | $7$ |
| Discriminant exponent $c$: | $7$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $-i$ |
| $\card{ \Gal(K/\Q_{ 3 }) }$: | $14$ |
| This field is Galois and abelian over $\Q_{3}.$ | |
| Visible slopes: | None |
Intermediate fields
| $\Q_{3}(\sqrt{3\cdot 2})$, 3.7.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 3.7.0.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{7} + 2 x^{2} + 1 \)
|
| Relative Eisenstein polynomial: |
\( x^{2} + 3 \)
$\ \in\Q_{3}(t)[x]$
|
Ramification polygon
Not computedInvariants of the Galois closure
| Galois group: | $C_{14}$ (as 14T1) |
| Inertia group: | Intransitive group isomorphic to $C_2$ |
| Wild inertia group: | $C_1$ |
| Unramified degree: | $7$ |
| Tame degree: | $2$ |
| Wild slopes: | None |
| Galois mean slope: | $1/2$ |
| Galois splitting model: | Not computed |