Defining polynomial
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\(x^{15} + 15 x^{12} + 30 x^{11} + 15 x^{10} + 750 x^{9} + 300 x^{8} + 450 x^{7} + 7425 x^{6} + 7575 x^{5} + 5250 x^{4} + 2500 x^{3} + 1875 x^{2} + 750 x + 125\)
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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: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $3$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[5/4]$ |
Intermediate fields
| 5.3.0.1, 5.5.5.4 |
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: |
\( x^{5} + \left(5 t^{2} + 10 t + 15\right) x^{2} + 10 x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 3$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_3\times F_5$ (as 15T8) |
| Inertia group: | Intransitive group isomorphic to $F_5$ |
| Wild inertia group: | $C_5$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4]$ |
| Galois mean slope: | $23/20$ |
| Galois splitting model: |
$x^{15} - 5 x^{14} + 30 x^{13} - 35 x^{12} + 30 x^{11} + 826 x^{10} - 3050 x^{9} + 3340 x^{8} - 2835 x^{7} - 84875 x^{6} + 64148 x^{5} + 161235 x^{4} - 11775 x^{3} - 58555 x^{2} + 54250 x - 679$
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