Defining polynomial
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\(x^{15} + 5 x^{14} + 15 x^{12} + 5 x^{11} + 20 x^{10} + 5 x^{5} + 5\)
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $25$ |
| 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: | $[23/12]$ |
Intermediate fields
| 5.3.2.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
| Unramified subfield: | $\Q_{5}$ |
| Relative Eisenstein polynomial: |
\( x^{15} + 5 x^{14} + 15 x^{12} + 5 x^{11} + 20 x^{10} + 5 x^{5} + 5 \)
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Ramification polygon
| Residual polynomials: | $3z + 4$,$z^{10} + 3z^{5} + 3$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[11, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5^2:(C_4\times S_3)$ (as 15T27) |
| Inertia group: | $C_5^2:C_{12}$ (as 15T19) |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $12$ |
| Wild slopes: | $[23/12, 23/12]$ |
| Galois mean slope: | $563/300$ |
| Galois splitting model: |
$x^{15} - 20 x^{13} - 80 x^{12} + 460 x^{11} + 376 x^{10} - 2500 x^{9} + 2520 x^{8} - 21195 x^{7} + 38840 x^{6} + 59792 x^{5} - 185000 x^{4} + 136070 x^{3} + 23280 x^{2} - 70560 x + 27648$
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