Defining polynomial
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$( x^{3} + 3 x + 3 )^{5} + \left(10 x + 10\right) ( x^{3} + 3 x + 3 )^{4} + 5$
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Invariants
| Base field: | $\Q_{5}$ |
| Degree $d$: | $15$ |
| Ramification index $e$: | $5$ |
| Residue field degree $f$: | $3$ |
| Discriminant exponent $c$: | $24$ |
| Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
| Root number: | $1$ |
| $\Aut(K/\Q_{5})$: | $C_1$ |
| This field is not Galois over $\Q_{5}.$ | |
| Visible Artin slopes: | $[2]$ |
| Visible Swan slopes: | $[1]$ |
| Means: | $\langle\frac{4}{5}\rangle$ |
| Rams: | $(1)$ |
| Jump set: | undefined |
| Roots of unity: | $124 = (5^{ 3 } - 1)$ |
Intermediate fields
| 5.3.1.0a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
| Unramified subfield: | 5.3.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{3} + 3 x + 3 \)
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| Relative Eisenstein polynomial: |
\( x^{5} + \left(5 t^{2} + 20\right) x^{4} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + (4 t^2 + 3 t + 2)$ |
| Associated inertia: | $4$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1500$ |
| Galois group: | $C_5^3:C_{12}$ (as 15T38) |
| Inertia group: | Intransitive group isomorphic to $C_5^3$ |
| Wild inertia group: | $C_5^3$ |
| Galois unramified degree: | $12$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2, 2]$ |
| Galois Swan slopes: | $[1,1,1]$ |
| Galois mean slope: | $1.984$ |
| Galois splitting model: |
$x^{15} - 90 x^{13} - 160 x^{12} + 4935 x^{11} + 10344 x^{10} - 138450 x^{9} - 431940 x^{8} + 2642510 x^{7} + 10032540 x^{6} - 37112344 x^{5} - 137222400 x^{4} + 373425015 x^{3} + 1141392300 x^{2} - 2340156640 x - 889176128$
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