$p$-adic field 2.2.8.40c1.12

• Label: 2.2.8.40c1.12
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(40\)
• Galois group: $C_2\wr S_3$ (as 16T765)

Defining polynomial

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

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

$\Q_{2}(\sqrt{5})$, 2.2.2.6a1.4, 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:	\( x^{8} + 4 x^{5} + 4 t x^{4} + 4 x^{2} + 8 t + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$384$
Galois group:	$C_2\wr S_3$ (as 16T765)
Inertia group:	Intransitive group isomorphic to $C_2^2\wr C_3$
Wild inertia group:	$C_2^6$
Galois unramified degree:	$2$
Galois tame degree:	$3$
Galois Artin slopes:	$[\frac{4}{3}, \frac{4}{3}, 2, \frac{8}{3}, \frac{8}{3}, 3]$
Galois Swan slopes:	$[\frac{1}{3},\frac{1}{3},1,\frac{5}{3},\frac{5}{3},2]$
Galois mean slope:	$2.6979166666666665$
Galois splitting model:	$x^{16} - 24 x^{14} + 76 x^{12} - 584 x^{10} + 7086 x^{8} - 20360 x^{6} + 6460 x^{4} - 600 x^{2} + 25$