Defining polynomial
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$( x^{5} + 2 x + 1 )^{3} + 6 x^{2} ( x^{5} + 2 x + 1 ) + 3$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3})$ |
| Root number: | $-i$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{3}{2}]$ |
| Visible Swan slopes: | $[\frac{1}{2}]$ |
| Means: | $\langle\frac{1}{3}\rangle$ |
| Rams: | $(\frac{1}{2})$ |
| Jump set: | undefined |
| Roots of unity: | $242 = (3^{ 5 } - 1)$ |
Intermediate fields
| 3.5.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 3.5.1.0a1.1 $\cong \Q_{3}(t)$ where $t$ is a root of
\( x^{5} + 2 x + 1 \)
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| Relative Eisenstein polynomial: |
\( x^{3} + \left(6 t^{4} + 3 t^{3} + 6 t^{2} + 6 t + 6\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^{2} + t)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2430$ |
| Galois group: | $C_3^5:C_{10}$ (as 15T44) |
| Inertia group: | Intransitive group isomorphic to $C_3^4:S_3$ |
| Wild inertia group: | $C_3^5$ |
| Galois unramified degree: | $5$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ |
| Galois mean slope: | $1.4958847736625513$ |
| Galois splitting model: |
$x^{15} + 1869 x^{13} - 17266 x^{12} + 926757 x^{11} - 16040025 x^{10} + 134482738 x^{9} - 3933727020 x^{8} + 23949206868 x^{7} + 19597433231 x^{6} + 2611645781625 x^{5} + 15129099314571 x^{4} - 381939000130630 x^{3} - 893963245309308 x^{2} + 10575767455709529 x + 22563615112703831$
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