Defining polynomial
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\(x^{15} - 60 x^{12} - 60 x^{11} + 15 x^{10} + 525 x^{9} + 2175 x^{8} + 825 x^{7} + 42025 x^{6} + 78450 x^{5} + 43875 x^{4} + 15000 x^{3} + 5625 x^{2} - 1500 x + 125\)
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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: | $\Q_{5}(\sqrt{5\cdot 2})$ |
| 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: |
\( x^{5} + \left(15 t^{2} + 10\right) x^{2} + \left(10 t^{2} + 5 t\right) x + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + 3t^{2} + 4t$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[1, 0]$ |
Invariants of the Galois closure
| Galois group: | $C_5^2:C_{12}$ (as 15T19) |
| Inertia group: | Intransitive group isomorphic to $C_5:F_5$ |
| Wild inertia group: | $C_5^2$ |
| Unramified degree: | $3$ |
| Tame degree: | $4$ |
| Wild slopes: | $[5/4, 5/4]$ |
| Galois mean slope: | $123/100$ |
| Galois splitting model: |
$x^{15} + 30 x^{13} - 415 x^{11} - 2760 x^{10} - 5950 x^{9} - 460 x^{8} + 55030 x^{7} + 227740 x^{6} + 592312 x^{5} + 1117400 x^{4} + 1629185 x^{3} + 1820580 x^{2} + 1428320 x + 817216$
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