Defining polynomial
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$( x^{3} + 3 x + 3 )^{5} + 10 x ( x^{3} + 3 x + 3 )^{4} + 5$
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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: | $\Q_{5}$ |
| 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: |
\( x^{5} + \left(5 t^{2} + 10\right) x^{4} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + (t^2 + 2 t + 1)$ |
| Associated inertia: | $2$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $150$ |
| Galois group: | $C_5^2:C_6$ (as 15T12) |
| Inertia group: | Intransitive group isomorphic to $C_5^2$ |
| Wild inertia group: | $C_5^2$ |
| Galois unramified degree: | $6$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2]$ |
| Galois Swan slopes: | $[1,1]$ |
| Galois mean slope: | $1.92$ |
| Galois splitting model: |
$x^{15} - 90 x^{13} - 30 x^{12} + 2715 x^{11} + 1578 x^{10} - 32400 x^{9} - 26160 x^{8} + 132810 x^{7} + 154270 x^{6} - 91470 x^{5} - 110400 x^{4} + 29145 x^{3} + 11520 x^{2} - 2700 x + 24$
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