Properties

Label 2.12.12.11
Base \(\Q_{2}\)
Degree \(12\)
e \(2\)
f \(6\)
c \(12\)
Galois group $A_4 \times C_2$ (as 12T7)

Related objects

Learn more about

Defining polynomial

\( x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57 \)

Invariants

Base field: $\Q_{2}$
Degree $d$ : $12$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $6$
Discriminant exponent $c$ : $12$
Discriminant root field: $\Q_{2}$
Root number: $-i$
$|\Aut(K/\Q_{ 2 })|$: $4$
This field is not Galois over $\Q_{2}$.

Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.0.1, 2.6.0.1, 2.6.6.1, 2.6.6.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} - x + 1 \)
Relative Eisenstein polynomial:$ x^{2} + 2 x + 2 t^{3} + 2 t^{2} + 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$A_4 \times C_2$ (as 12T7)
Inertia group:Intransitive group isomorphic to $C_2^2$
Unramified degree:$6$
Tame degree:$1$
Wild slopes:[2, 2]
Galois mean slope:$3/2$
Galois splitting model:\( x^{12} + 6 x^{10} + 15 x^{8} + 19 x^{6} + 12 x^{4} + 3 x^{2} + 1 \)