Defining polynomial
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$( x^{5} + 2 x + 1 )^{3} + \left(9 x^{3} + 9 x\right) ( x^{5} + 2 x + 1 ) + 3$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $25$ |
| Discriminant root field: | $\Q_{3}(\sqrt{3\cdot 2})$ |
| Root number: | $i$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[\frac{5}{2}]$ |
| Visible Swan slopes: | $[\frac{3}{2}]$ |
| Means: | $\langle1\rangle$ |
| Rams: | $(\frac{3}{2})$ |
| 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: |
\( x^{3} + \left(9 t^{3} + 9 t\right) x + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (t^4 + t^3 + t^2 + 1)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 0]$ |
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: | $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{3}{2}, \frac{5}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{3}{2}]$ |
| Galois mean slope: | $2.162551440329218$ |
| Galois splitting model: |
$x^{15} - 9 x^{12} - 126 x^{9} + 405 x^{6} + 81 x^{3} - 243$
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