Defining polynomial
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\(x^{15} + 20 x^{4} + 5\)
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $15$ |
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| Ramification index $e$: | $15$ |
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| Residue field degree $f$: | $1$ |
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| Discriminant exponent $c$: | $18$ |
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| Discriminant root field: | $\Q_{5}$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{4}{3}]$ | |
| Visible Swan slopes: | $[\frac{1}{3}]$ | |
| Means: | $\langle\frac{4}{15}\rangle$ | |
| Rams: | $(1)$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
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Intermediate fields
| 5.1.3.2a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | $\Q_{5}$ |
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| Relative Eisenstein polynomial: |
\( x^{15} + 20 x^{4} + 5 \)
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Ramification polygon
| Residual polynomials: | $z^{10} + 3 z^5 + 3$,$3 z^4 + 4$ |
| Associated inertia: | $2$,$4$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $300$ |
| Galois group: | $C_5^2:C_3:C_4$ (as 15T17) |
| Inertia group: | $C_5^2:C_3$ (as 15T9) |
| Wild inertia group: | $C_5^2$ |
| Galois unramified degree: | $4$ |
| Galois tame degree: | $3$ |
| Galois Artin slopes: | $[\frac{4}{3}, \frac{4}{3}]$ |
| Galois Swan slopes: | $[\frac{1}{3},\frac{1}{3}]$ |
| Galois mean slope: | $1.3066666666666666$ |
| Galois splitting model: |
$x^{15} - 5 x^{14} + 5 x^{13} - 25 x^{12} + 210 x^{11} - 586 x^{10} + 1430 x^{9} - 3400 x^{8} + 1600 x^{7} + 4860 x^{6} + 3352 x^{5} - 5520 x^{4} - 15880 x^{3} - 9120 x^{2} + 432$
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