$p$-adic field 2.2.8.32d5.2

• Label: 2.2.8.32d5.2
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(32\)
• Galois group: $C_2^6.A_4\wr C_2$ (as 16T1783)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{2} + x + 1 ) + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$32$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_1$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, \frac{7}{3}, \frac{7}{3}]$
Visible Swan slopes:	$[1,\frac{4}{3},\frac{4}{3}]$
Means:	$\langle\frac{1}{2}, \frac{11}{12}, \frac{9}{8}\rangle$
Rams:	$(1, \frac{5}{3}, \frac{5}{3})$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

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

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

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$18432$
Galois group:	$C_2^6.A_4\wr C_2$ (as 16T1783)
Inertia group:	intransitive group not computed
Wild inertia group:	not computed
Galois unramified degree:	$6$
Galois tame degree:	$3$
Galois Artin slopes:	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2, \frac{7}{3}, \frac{7}{3}, \frac{7}{3}, \frac{7}{3}]$
Galois Swan slopes:	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},1,1,\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]$
Galois mean slope:	$2.3014322916666665$
Galois splitting model:	$x^{16} + 12 x^{14} - 32 x^{13} + 240 x^{12} - 720 x^{11} + 2568 x^{10} - 5472 x^{9} + 8172 x^{8} - 7040 x^{7} + 9360 x^{6} - 34944 x^{5} + 59056 x^{4} - 58368 x^{3} + 43200 x^{2} - 12096 x + 1296$