$p$-adic field 2.2.8.48c13.264

• Label: 2.2.8.48c13.264
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(48\)
• Galois group: $C_2^2.C_2\wr C_4$ (as 16T681)

Defining polynomial

$( x^{2} + x + 1 )^{8} + \left(8 x + 8\right) ( x^{2} + x + 1 )^{7} + 6 ( x^{2} + x + 1 )^{6} + \left(12 x + 14\right) ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + \left(8 x + 4\right) ( x^{2} + x + 1 )^{3} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{2} + 8 x ( x^{2} + x + 1 ) + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$48$
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:	$[2, 3, 4]$
Visible Swan slopes:	$[1,2,3]$
Means:	$\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}\rangle$
Rams:	$(1, 3, 7)$
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, 2.2.4.16b5.5

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

Ramification polygon

Residual polynomials:	$z^4 + t$,$t z^2 + (t + 1)$,$(t + 1) z + t$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[17, 10, 4, 0]$

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:	$[2, 2, 2, 3, \frac{7}{2}, 4]$
Galois Swan slopes:	$[1,1,1,2,\frac{5}{2},3]$
Galois mean slope:	$3.46875$
Galois splitting model:	not computed