Defining polynomial
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$( x^{3} + 3 x + 3 )^{5} + \left(15 x^{2} + 20 x + 10\right) ( x^{3} + 3 x + 3 )^{2} + 5$
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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 $c$: | $18$ |
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| 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: | $[\frac{3}{2}]$ | |
| Visible Swan slopes: | $[\frac{1}{2}]$ | |
| Means: | $\langle\frac{2}{5}\rangle$ | |
| Rams: | $(\frac{1}{2})$ | |
| 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: |
\( x^{5} + \left(15 t^{2} + 20 t + 10\right) x^{2} + 5 \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
| Residual polynomials: | $z^2 + (t^2 + 2 t + 2)$ |
| Associated inertia: | $1$ |
| Indices of inseparability: | $[2, 0]$ |
Invariants of the Galois closure
| Galois degree: | $750$ |
| Galois group: | $C_5^3:C_6$ (as 15T30) |
| Inertia group: | Intransitive group isomorphic to $C_5^3:C_2$ |
| Wild inertia group: | $C_5^3$ |
| Galois unramified degree: | $3$ |
| Galois tame degree: | $2$ |
| Galois Artin slopes: | $[\frac{3}{2}, \frac{3}{2}, \frac{3}{2}]$ |
| Galois Swan slopes: | $[\frac{1}{2},\frac{1}{2},\frac{1}{2}]$ |
| Galois mean slope: | $1.492$ |
| Galois splitting model: |
$x^{15} - 6765 x^{13} - 6765 x^{12} + 15606855 x^{11} + 27179966 x^{10} - 14355025575 x^{9} - 38124466435 x^{8} + 4795156200890 x^{7} + 26949193437925 x^{6} - 443903213159681 x^{5} - 2330684245703275 x^{4} + 17928124882860555 x^{3} + 59203909199756310 x^{2} - 285419451834333850 x - 217857881265284777$
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