Defining polynomial
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\(x^{15} + 5 x^{14} + 10 x^{13} + 25 x + 5\)
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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 $c$: | $27$ |
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| Discriminant root field: | $\Q_{5}(\sqrt{5})$ | |
| Root number: | $-1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{25}{12}]$ | |
| Visible Swan slopes: | $[\frac{13}{12}]$ | |
| Means: | $\langle\frac{13}{15}\rangle$ | |
| Rams: | $(\frac{13}{4})$ | |
| 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: |
\( x^{15} + 5 x^{14} + 10 x^{13} + 25 x + 5 \)
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Ramification polygon
| Residual polynomials: | $z^{10} + 3 z^5 + 3$,$3 z + 4$ |
| Associated inertia: | $2$,$1$ |
| Indices of inseparability: | $[13, 0]$ |
Invariants of the Galois closure
| Galois degree: | $3000$ |
| Galois group: | $C_5^3:(C_4\times S_3)$ (as 15T49) |
| Inertia group: | $C_5^3:C_{12}$ (as 15T38) |
| Wild inertia group: | $C_5^3$ |
| Galois unramified degree: | $2$ |
| Galois tame degree: | $12$ |
| Galois Artin slopes: | $[\frac{7}{4}, \frac{25}{12}, \frac{25}{12}]$ |
| Galois Swan slopes: | $[\frac{3}{4},\frac{13}{12},\frac{13}{12}]$ |
| Galois mean slope: | $2.0633333333333335$ |
| Galois splitting model: |
$x^{15} - 70 x^{13} - 20 x^{12} + 1435 x^{11} + 332 x^{10} + 19000 x^{9} - 156160 x^{8} - 46360 x^{7} + 656080 x^{6} + 1844328 x^{5} + 8722000 x^{4} - 46211620 x^{3} + 645680 x^{2} - 127424640 x + 698668544$
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