Defining polynomial
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$( x^{3} + 3 x + 3 )^{5} + \left(5 x^{2} + 20 x + 5\right) ( x^{3} + 3 x + 3 )^{3} + 5$
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Invariants
| Base field: | $\Q_{5}$ |
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| Degree $d$: | $15$ |
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| Ramification index $e$: | $5$ |
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| Residue field degree $f$: | $3$ |
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| Discriminant exponent $c$: | $21$ |
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| Discriminant root field: | $\Q_{5}(\sqrt{5\cdot 2})$ | |
| Root number: | $1$ | |
| $\Aut(K/\Q_{5})$: | $C_1$ | |
| This field is not Galois over $\Q_{5}.$ | ||
| Visible Artin slopes: | $[\frac{7}{4}]$ | |
| Visible Swan slopes: | $[\frac{3}{4}]$ | |
| Means: | $\langle\frac{3}{5}\rangle$ | |
| Rams: | $(\frac{3}{4})$ | |
| Jump set: | undefined | |
| Roots of unity: | $124 = (5^{ 3 } - 1)$ |
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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: |
\( x^{5} + \left(15 t^{2} + 15 t\right) x^{3} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z + (3 t + 3)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[3, 0]$ |
Invariants of the Galois closure
| Galois degree: | $1500$ |
| Galois group: | $C_5^3:C_{12}$ (as 15T38) |
| Inertia group: | Intransitive group isomorphic to $C_5^3:C_4$ |
| Wild inertia group: | $C_5^3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $4$ |
| Galois Artin slopes: | $[\frac{7}{4}, \frac{7}{4}, \frac{7}{4}]$ |
| Galois Swan slopes: | $[\frac{3}{4},\frac{3}{4},\frac{3}{4}]$ |
| Galois mean slope: | $1.742$ |
| Galois splitting model: |
$x^{15} + 70 x^{13} - 560 x^{12} + 2345 x^{11} - 18088 x^{10} + 360850 x^{9} - 2815540 x^{8} + 5628070 x^{7} - 90677300 x^{6} + 221297832 x^{5} - 206920000 x^{4} + 3495058175 x^{3} - 12585885900 x^{2} + 14672069440 x - 5032433728$
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