Defining polynomial
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\(x^{15} + 5 x^{14} + 10 x^{10} + 10 x^{5} + 100 x + 80\)
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification exponent $e$: | $15$ |
| Residue field degree $f$: | $1$ |
| Discriminant exponent $c$: | $28$ |
| Discriminant root field: | $\Q_{5}$ |
| Root number: | $1$ |
| $\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible slopes: | $[13/6]$ |
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: |
\( x^{15} + 5 x^{14} + 10 x^{10} + 10 x^{5} + 100 x + 80 \)
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Ramification polygon
| Residual polynomials: | $3z^{2} + 1$,$z^{10} + 3z^{5} + 3$ |
| Associated inertia: | $2$,$2$ |
| Indices of inseparability: | $[14, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5^2:C_3:C_4$ (as 15T17) |
| Inertia group: | $C_5^2:C_6$ (as 15T12) |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $2$ |
| Tame degree: | $6$ |
| Wild slopes: | $[13/6, 13/6]$ |
| Galois mean slope: | $317/150$ |
| Galois splitting model: |
$x^{15} - 70 x^{13} - 5 x^{12} + 1955 x^{11} + 446 x^{10} - 34975 x^{9} + 29870 x^{8} + 328405 x^{7} - 1193505 x^{6} + 2154667 x^{5} - 809575 x^{4} - 2574905 x^{3} + 1456930 x^{2} + 1038820 x + 158728$
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