Properties

Label 5.4.3.8a1.1
Base \(\Q_{5}\)
Degree \(12\)
e \(3\)
f \(4\)
c \(8\)
Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

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Defining polynomial

$( x^{4} + 4 x^{2} + 4 x + 2 )^{3} + 5 x$ Copy content Toggle raw display

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 \( x^{4} + 4 x^{2} + 4 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 5 t \) $\ \in\Q_{5}(t)[x]$ Copy content Toggle raw display

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: $x^{12} - 456 x^{10} - 1260 x^{9} + 74196 x^{8} + 406980 x^{7} - 4632334 x^{6} - 40941180 x^{5} + 35890293 x^{4} + 1261978200 x^{3} + 3880645326 x^{2} + 1156875300 x - 6917034473$ Copy content Toggle raw display