Defining polynomial
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\(x^{15} + 5 x^{6} + 5\)
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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 $c$: | $20$ |
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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: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{2}{5}\rangle$ | |
| Rams: | $(\frac{3}{2})$ | |
| Jump set: | undefined | |
| Roots of unity: | $4 = (5 - 1)$ |
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Intermediate fields
| 5.1.3.2a1.1, 5.1.5.6a2.1 |
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: |
\( x^{15} + 5 x^{6} + 5 \)
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Ramification polygon
| Residual polynomials: | $z^{10} + 3 z^5 + 3$,$3 z^2 + 4$ |
| Associated inertia: | $2$,$2$ |
| Indices of inseparability: | $[6, 0]$ |
Invariants of the Galois closure
| Galois degree: | $60$ |
| Galois group: | $C_{15}:C_4$ (as 15T6) |
| Inertia group: | $C_3\times D_5$ (as 15T3) |
| Wild inertia group: | $C_5$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $6$ |
| Galois Artin slopes: | $[\frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2}]$ |
| Galois mean slope: | $1.3666666666666667$ |
| Galois splitting model: |
$x^{15} - 5 x^{14} + 60 x^{12} - 115 x^{11} + 293 x^{10} - 145 x^{9} + 1245 x^{8} - 4425 x^{7} + 6125 x^{6} - 1967 x^{5} - 16865 x^{4} + 37570 x^{3} - 8060 x^{2} - 8385 x + 8411$
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