$p$-adic field 2.2.8.24b4.1

• Label: 2.2.8.24b4.1
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(24\)
• Galois group: $\SL(2,3)\wr C_2$ (as 16T1291)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{3} + 2 x ( x^{2} + x + 1 )^{2} + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$24$
Discriminant root field:	$\Q_{2}$
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

$\Q_{2}(\sqrt{5})$, 2.2.4.8a2.1

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:	\( x^{8} + 2 t x^{5} + 2 t x^{2} + 6 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$1152$
Galois group:	$\SL(2,3)\wr C_2$ (as 16T1291)
Inertia group:	Intransitive group isomorphic to $Q_8:\SL(2,3)$
Wild inertia group:	$Q_8^2$
Galois unramified degree:	$6$
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:	$x^{16} - 12 x^{12} - 20 x^{10} + 156 x^{8} - 528 x^{6} + 1216 x^{4} - 768 x^{2} + 144$