Defining polynomial
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$( x^{3} + 3 x + 3 )^{5} + 15 x ( x^{3} + 3 x + 3 )^{4} + 5$
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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: | $1$ |
| $\Aut(K/\Q_{5})$: | $C_5$ |
| 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: |
\( x^{5} + \left(5 t^{2} + 10 t + 10\right) x^{4} + 50 t + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^4 + (4 t^2 + 3 t + 4)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[4, 0]$ |
Invariants of the Galois closure
| Galois degree: | $375$ |
| Galois group: | $C_5\wr C_3$ (as 15T25) |
| Inertia group: | Intransitive group isomorphic to $C_5^2$ |
| Wild inertia group: | $C_5^2$ |
| Galois unramified degree: | $15$ |
| Galois tame degree: | $1$ |
| Galois Artin slopes: | $[2, 2]$ |
| Galois Swan slopes: | $[1,1]$ |
| Galois mean slope: | $1.92$ |
| Galois splitting model: |
$x^{15} - 4180 x^{13} - 76010 x^{12} + 4707120 x^{11} + 171319500 x^{10} + 380632725 x^{9} - 64599686305 x^{8} - 1080227091020 x^{7} - 1302083854390 x^{6} + 144036618452991 x^{5} + 1827573911298300 x^{4} + 10043234975446035 x^{3} + 24347243897070930 x^{2} + 9340304205459535 x - 37074934376752679$
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