Defining polynomial
\(x^{15} - 30 x^{12} + 15 x^{10} + 2025 x^{9} + 2550 x^{8} + 600 x^{7} + 9625 x^{6} + 54825 x^{5} + 76125 x^{4} + 35750 x^{3} + 3750 x^{2} + 125\)
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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: | $\Q_{5}(\sqrt{5})$ |
Root number: | $-1$ |
$\card{ \Aut(K/\Q_{ 5 }) }$: | $1$ |
This field is not Galois over $\Q_{5}.$ | |
Visible slopes: | $[5/4]$ |
Intermediate fields
5.3.0.1 |
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: |
\( x^{5} + \left(10 t^{2} + 15 t + 10\right) x^{2} + \left(10 t^{2} + 10 t + 20\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
Residual polynomials: | $z + 3t^{2} + 3t + 1$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
Galois group: | $C_5^3:C_{12}$ (as 15T38) |
Inertia group: | Intransitive group isomorphic to $C_5^3:C_4$ |
Wild inertia group: | $C_5^3$ |
Unramified degree: | $3$ |
Tame degree: | $4$ |
Wild slopes: | $[5/4, 5/4, 5/4]$ |
Galois mean slope: | $623/500$ |
Galois splitting model: |
$x^{15} + 5 x^{13} - 25 x^{11} - 2 x^{10} - 100 x^{9} + 180 x^{8} + 25 x^{7} + 560 x^{6} + 102 x^{5} + 450 x^{4} - 495 x^{3} - 100 x^{2} - 300 x + 232$
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