$p$-adic field 2.2.8.40c2.17

• Label: 2.2.8.40c2.17
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(40\)
• Galois group: $C_2^4.S_4$ (as 16T750)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 4 x ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{2} + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$40$
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:	$[\frac{8}{3}, \frac{8}{3}, 3]$
Visible Swan slopes:	$[\frac{5}{3},\frac{5}{3},2]$
Means:	$\langle\frac{5}{6}, \frac{5}{4}, \frac{13}{8}\rangle$
Rams:	$(\frac{5}{3}, \frac{5}{3}, 3)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.4.16a1.2

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:	\( x^{8} + \left(4 t + 4\right) x^{7} + 4 t x^{5} + 4 x^{4} + 4 x^{2} + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^2 + 1$,$z + t$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[13, 10, 8, 0]$

Invariants of the Galois closure

Galois degree:	$384$
Galois group:	$C_2^4.S_4$ (as 16T750)
Inertia group:	Intransitive group isomorphic to $C_2^4:A_4$
Wild inertia group:	$D_4:C_2^3$
Galois unramified degree:	$2$
Galois tame degree:	$3$
Galois Artin slopes:	$[\frac{4}{3}, \frac{4}{3}, \frac{8}{3}, \frac{8}{3}, 3, 3]$
Galois Swan slopes:	$[\frac{1}{3},\frac{1}{3},\frac{5}{3},\frac{5}{3},2,2]$
Galois mean slope:	$2.8229166666666665$
Galois splitting model:	$x^{16} - 144 x^{14} + 7704 x^{12} - 195048 x^{10} + 2481894 x^{8} - 15829776 x^{6} + 49187736 x^{4} - 67287672 x^{2} + 28100601$