$p$-adic field 2.2.8.40b2.124

• Label: 2.2.8.40b2.124
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(40\)
• Galois group: $C_2^5:C_4$ (as 16T227)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 4 x ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 4 x + 6$

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:	$-1$
$\Aut(K/\Q_{2})$:	$C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 2, \frac{7}{2}]$
Visible Swan slopes:	$[1,1,\frac{5}{2}]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{13}{8}\rangle$
Rams:	$(1, 1, 7)$
Jump set:	$[1, 3, 6, 16]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.4a1.2, 2.2.2.4a2.2, 2.2.2.4a2.1, 2.2.4.12a1.2, 2.2.4.18a2.6, 2.2.4.18a2.16

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:	\( x^{8} + 4 t x^{7} + 2 x^{6} + \left(4 t + 4\right) x^{5} + 8 x^{4} + 8 t x^{3} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^6 + 1$,$z + t$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[13, 6, 6, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_2^5:C_4$ (as 16T227)
Inertia group:	Intransitive group isomorphic to $C_2^2\wr C_2$
Wild inertia group:	$C_2^2\wr C_2$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2}]$
Galois mean slope:	$3.1875$
Galois splitting model:	$x^{16} - 10 x^{14} + 50 x^{12} - 150 x^{10} + 230 x^{8} - 150 x^{6} + 50 x^{4} - 50 x^{2} + 25$