Properties

Label 2.3.4.33a1.187
Base \(\Q_{2}\)
Degree \(12\)
e \(4\)
f \(3\)
c \(33\)
Galois group $C_2^4:C_{12}$ (as 12T105)

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Defining polynomial

$( x^{3} + x + 1 )^{4} + 8 ( x^{3} + x + 1 )^{3} + 4 ( x^{3} + x + 1 )^{2} + 8 x^{2} ( x^{3} + x + 1 ) + 2$ Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $12$
Ramification index $e$: $4$
Residue field degree $f$: $3$
Discriminant exponent $c$: $33$
Discriminant root field: $\Q_{2}(\sqrt{2})$
Root number: $1$
$\Aut(K/\Q_{2})$: $C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:$[3, 4]$
Visible Swan slopes:$[2,3]$
Means:$\langle1, 2\rangle$
Rams:$(2, 4)$
Jump set:$[1, 3, 7]$
Roots of unity:$14 = (2^{ 3 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{2})$, 2.3.1.0a1.1, 2.3.2.9a1.5

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

Canonical tower

Unramified subfield:2.3.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{3} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{4} + 8 x^{3} + 4 x^{2} + 8 t x + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^2 + (t + 1)$,$(t + 1) z + (t^2 + 1)$
Associated inertia:$1$,$1$
Indices of inseparability:$[8, 4, 0]$

Invariants of the Galois closure

Galois degree: $192$
Galois group: $C_2^4:C_{12}$ (as 12T105)
Inertia group: Intransitive group isomorphic to $C_2^4:C_4$
Wild inertia group: $C_2^4:C_4$
Galois unramified degree: $3$
Galois tame degree: $1$
Galois Artin slopes: $[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]$
Galois Swan slopes: $[1,1,2,\frac{5}{2},\frac{5}{2},3]$
Galois mean slope: $3.59375$
Galois splitting model:$x^{12} + 12 x^{10} + 40 x^{8} - 172 x^{4} - 176 x^{2} + 8$