Defining polynomial
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\(x^{15} + 20 x^{5} + 10 x^{4} + 5\)
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $18$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $5$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[4/3]$ |
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: |
\( x^{15} + 20 x^{5} + 10 x^{4} + 5 \)
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Ramification polygon
| Residual polynomials: | $3z^{4} + 2$,$z^{10} + 3z^{5} + 3$ |
| Associated inertia: | $1$,$2$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5\wr S_3$ (as 15T32) |
| Inertia group: | $C_5^2:C_3$ (as 15T9) |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $10$ |
| Tame degree: | $3$ |
| Wild slopes: | $[4/3, 4/3]$ |
| Galois mean slope: | $98/75$ |
| Galois splitting model: |
$x^{15} - 5 x^{14} + 15 x^{13} - 60 x^{12} + 145 x^{11} - 242 x^{10} + 440 x^{9} - 220 x^{8} - 945 x^{7} + 1140 x^{6} + 288 x^{5} - 55 x^{4} - 485 x^{3} + 540 x^{2} + 595 x - 369$
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