$p$-adic field 2.2.8.32c10.4

• Label: 2.2.8.32c10.4
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(32\)
• Galois group: $C_2^2.C_2\wr C_4$ (as 16T681)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 2\right) ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$

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:	$[2, 2, \frac{5}{2}]$
Visible Swan slopes:	$[1,1,\frac{3}{2}]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{9}{8}\rangle$
Rams:	$(1, 1, 3)$
Jump set:	$[1, 2, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.4a2.1, 2.2.4.12a4.1

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

Ramification polygon

Residual polynomials:	$z^6 + z^2 + t$,$t z + (t + 1)$
Associated inertia:	$2$,$1$
Indices of inseparability:	$[9, 6, 4, 0]$

Invariants of the Galois closure

Galois degree:	$256$
Galois group:	$C_2^2.C_2\wr C_4$ (as 16T681)
Inertia group:	not computed
Wild inertia group:	not computed
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, \frac{9}{4}, \frac{5}{2}]$
Galois Swan slopes:	$[1,1,1,1,\frac{5}{4},\frac{3}{2}]$
Galois mean slope:	$2.28125$
Galois splitting model:	not computed