Defining polynomial
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\(x^{15} - 225 x^{12} - 135 x^{10} + 41250 x^{9} + 97500 x^{8} + 147750 x^{7} + 5776875 x^{6} + 73575 x^{5} + 9300000 x^{4} - 5512500 x^{3} - 193125 x^{2} - 5475000 x + 496375\)
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $27$ |
| Discriminant root field: | $\Q_{5}(\sqrt{5})$ |
| Root number: | $-1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $3$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[9/4]$ |
Intermediate fields
| 5.3.0.1, 5.5.9.3 |
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: |
\( x^{5} + \left(75 t^{2} + 50 t + 75\right) x^{2} + 100 t x + 50 t^{2} + 100 t + 55 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 4$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[5, 0]$ |
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: | $[9/4]$ |
| Galois mean slope: | $39/20$ |
| Galois splitting model: |
$x^{15} - 5 x^{14} + 35 x^{12} - 35 x^{11} + 149 x^{10} - 660 x^{9} + 4095 x^{8} - 10630 x^{7} + 22400 x^{6} - 9192 x^{5} - 20870 x^{4} + 321565 x^{3} - 246365 x^{2} + 409990 x + 405041$
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