$p$-adic field 2.2.8.24b2.2

• Label: 2.2.8.24b2.2
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(24\)
• Galois group: $C_2^4.\GL(2,3)$ (as 16T1069)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{2} + 6$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$24$
Discriminant root field:	$\Q_{2}(\sqrt{5})$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[\frac{4}{3}, \frac{4}{3}, 2]$
Visible Swan slopes:	$[\frac{1}{3},\frac{1}{3},1]$
Means:	$\langle\frac{1}{6}, \frac{1}{4}, \frac{5}{8}\rangle$
Rams:	$(\frac{1}{3}, \frac{1}{3}, 3)$
Jump set:	$[1, 2, 5, 13]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.1.4.4a1.1, 2.2.4.8a1.1

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

Canonical tower

Unramified subfield:	$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{8} + \left(2 t + 2\right) x^{7} + 2 t x^{5} + 2 x^{2} + 6 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^2 + 1$,$z + t$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[5, 2, 2, 0]$

Invariants of the Galois closure

Galois degree:	$768$
Galois group:	$C_2^4.\GL(2,3)$ (as 16T1069)
Inertia group:	not computed
Wild inertia group:	not computed
Galois unramified degree:	$4$
Galois tame degree:	$3$
Galois Artin slopes:	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]$
Galois Swan slopes:	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1]$
Galois mean slope:	$1.8229166666666667$
Galois splitting model:	not computed