Defining polynomial
$( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5 x$
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Invariants
Base field: | $\Q_{5}$ |
Degree $d$: | $12$ |
Ramification index $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{5}(\sqrt{2})$ |
Root number: | $1$ |
$\Aut(K/\Q_{5})$: | $C_6$ |
This field is not Galois over $\Q_{5}.$ | |
Visible Artin slopes: | $[\ ]$ |
Visible Swan slopes: | $[]$ |
Means: | $\langle\ \rangle$ |
Rams: | $(\ )$ |
Jump set: | undefined |
Roots of unity: | $624 = (5^{ 4 } - 1)$ |
Intermediate fields
$\Q_{5}(\sqrt{2})$, 5.4.1.0a1.1, 5.2.3.4a1.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Canonical tower
Unramified subfield: | 5.4.1.0a1.1 $\cong \Q_{5}(t)$ where $t$ is a root of
\( x^{4} + 4 x^{2} + 4 x + 2 \)
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Relative Eisenstein polynomial: |
\( x^{3} + 5 t \)
$\ \in\Q_{5}(t)[x]$
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Ramification polygon
Residual polynomials: | $z^{2} + 3 z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois degree: | $36$ |
Galois group: | $C_3:C_{12}$ (as 12T19) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Galois unramified degree: | $12$ |
Galois tame degree: | $3$ |
Galois Artin slopes: | $[\ ]$ |
Galois Swan slopes: | $[]$ |
Galois mean slope: | $0.6666666666666666$ |
Galois splitting model: |
$x^{12} - 456 x^{10} - 1260 x^{9} + 74196 x^{8} + 406980 x^{7} - 4632334 x^{6} - 40941180 x^{5} + 35890293 x^{4} + 1261978200 x^{3} + 3880645326 x^{2} + 1156875300 x - 6917034473$
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