Defining polynomial
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\(x^{15} - 90 x^{12} - 15 x^{11} + 15 x^{10} + 3675 x^{9} + 3825 x^{8} + 450 x^{7} + 6975 x^{6} - 8550 x^{5} + 7125 x^{4} + 14000 x^{3} + 4500 x^{2} - 375 x + 125\)
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $15$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[5/4]$ |
Intermediate fields
| 5.3.0.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{3} + 3 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{5} + \left(15 t^{2} + 10 t\right) x^{2} + \left(5 t^{2} + 15 t + 5\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 4t^{2} + 2t + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5^3:C_{12}$ (as 15T38) |
| Inertia group: | Intransitive group isomorphic to $C_5^3:C_4$ |
| Wild inertia group: | $C_5^3$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4, 5/4, 5/4]$ |
| Galois mean slope: | $623/500$ |
| Galois splitting model: |
$x^{15} + 10 x^{13} + 5 x^{11} - 4 x^{10} - 100 x^{9} + 20 x^{8} - 125 x^{7} + 140 x^{6} + 121 x^{5} + 200 x^{4} + 65 x^{3} - 50 x^{2} - 40 x + 8$
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