Defining polynomial
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$( x^{5} + 2 x + 1 )^{3} + \left(3 x^{2} + 3 x\right) ( x^{5} + 2 x + 1 )^{2} + 3$
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Invariants
| Base field: | $\Q_{3}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $3$ |
| Residue field degree $f$: | $5$ |
| Discriminant exponent $c$: | $20$ |
| Discriminant root field: | $\Q_{3}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{3})$: | $C_1$ |
| This field is not Galois over $\Q_{3}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{2}{3}\rangle$ |
| Rams: | $(1)$ |
| 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: |
\( x^{3} + \left(6 t^{3} + 3 t\right) x^{2} + 3 \)
$\ \in\Q_{3}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (2 t^4 + t^2 + 1)$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $2430$ |
| Galois group: | $C_3^5:C_{10}$ (as 15T44) |
| Inertia group: | Intransitive group isomorphic to $C_3^5$ |
| Wild inertia group: | $C_3^5$ |
| Galois unramified degree: | $10$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2, 2, 2]$ |
| Galois Swan slopes: | $[1,1,1,1,1]$ |
| Galois mean slope: | $1.991769547325103$ |
| Galois splitting model: |
$x^{15} + 15 x^{13} - 22 x^{12} - 9 x^{11} - 330 x^{10} - 621 x^{9} + 1494 x^{7} + 9284 x^{6} + 21825 x^{5} + 4026 x^{4} - 62524 x^{3} - 95832 x^{2} - 46464 x - 3872$
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