$p$-adic field 2.2.8.32b1.55

• Label: 2.2.8.32b1.55
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(32\)
• Galois group: $C_2^4.\GL(2,3)$ (as 16T1069)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{7} + 4 ( x^{2} + x + 1 )^{3} + 2 ( x^{2} + x + 1 )^{2} + 4 ( x^{2} + x + 1 ) + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$32$
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:	$[\frac{4}{3}, \frac{4}{3}, 3]$
Visible Swan slopes:	$[\frac{1}{3},\frac{1}{3},2]$
Means:	$\langle\frac{1}{6}, \frac{1}{4}, \frac{9}{8}\rangle$
Rams:	$(\frac{1}{3}, \frac{1}{3}, 7)$
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:	\( x^{8} + 4 t x^{7} + 4 x^{3} + 2 x^{2} + 4 x + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[9, 2, 2, 0]$

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:	$2$
Galois tame degree:	$3$
Galois Artin slopes:	$[\frac{4}{3}, \frac{4}{3}, \frac{3}{2}, 2, \frac{7}{3}, \frac{7}{3}, 3]$
Galois Swan slopes:	$[\frac{1}{3},\frac{1}{3},\frac{1}{2},1,\frac{4}{3},\frac{4}{3},2]$
Galois mean slope:	$2.5833333333333335$
Galois splitting model:	not computed