$p$-adic field 2.2.8.44c2.32

• Label: 2.2.8.44c2.32
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(44\)
• Galois group: $C_2^6.A_4\wr C_2$ (as 16T1783)

Defining polynomial

$( x^{2} + x + 1 )^{8} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{7} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 x ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 ) + 8 x + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$44$
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:	$[3, \frac{19}{6}, \frac{19}{6}]$
Visible Swan slopes:	$[2,\frac{13}{6},\frac{13}{6}]$
Means:	$\langle1, \frac{19}{12}, \frac{15}{8}\rangle$
Rams:	$(2, \frac{7}{3}, \frac{7}{3})$
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

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

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z + t$
Associated inertia:	$1$,$1$
Indices of inseparability:	$[15, 14, 8, 0]$

Invariants of the Galois closure

Galois degree:	$18432$
Galois group:	$C_2^6.A_4\wr C_2$ (as 16T1783)
Inertia group:	not computed
Wild inertia group:	not computed
Galois unramified degree:	not computed
Galois tame degree:	not computed
Galois Artin slopes:	not computed
Galois Swan slopes:	not computed
Galois mean slope:	not computed
Galois splitting model:	not computed