$p$-adic field 2.6.2.12a13.2

• Label: 2.6.2.12a13.2
• Base: \(\Q_{2}\)
• Degree: \(12\)
• e: \(2\)
• f: \(6\)
• c: \(12\)
• Galois group: $C_2\wr C_6$ (as 12T134)

Defining polynomial

$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$12$
Ramification index $e$:	$2$
Residue field degree $f$:	$6$
Discriminant exponent $c$:	$12$
Discriminant root field:	$\Q_{2}(\sqrt{-1})$
Root number:	$-i$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2]$
Visible Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(1)$
Jump set:	$[1, 3]$
Roots of unity:	$126 = (2^{ 6 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.3.1.0a1.1, 2.6.1.0a1.1

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

Canonical tower

Unramified subfield:	2.6.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} + x^{4} + x^{3} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{2} + \left(2 t^{3} + 2 t^{2} + 2\right) x + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z + (t^5 + t^4)$
Associated inertia:	$1$
Indices of inseparability:	$[1, 0]$

Invariants of the Galois closure

Galois degree:	$384$
Galois group:	$C_2\wr C_6$ (as 12T134)
Inertia group:	Intransitive group isomorphic to $C_2^6$
Wild inertia group:	$C_2^6$
Galois unramified degree:	$6$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, 2, 2]$
Galois Swan slopes:	$[1,1,1,1,1,1]$
Galois mean slope:	$1.96875$
Galois splitting model:	$x^{12} + 4 x^{10} - 2 x^{8} - 22 x^{6} - 24 x^{4} - 9 x^{2} - 1$