$p$-adic field 2.2.8.62a1.1320

• Label: 2.2.8.62a1.1320
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(62\)
• Galois group: $C_8.C_4$ (as 16T49)

Defining polynomial

$( x^{2} + x + 1 )^{8} + \left(16 x + 16\right) ( x^{2} + x + 1 )^{7} + 16 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{2} + 16 x + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$62$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_8$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, 4, 5]$
Visible Swan slopes:	$[2,3,4]$
Means:	$\langle1, 2, 3\rangle$
Rams:	$(2, 4, 8)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.2.2.6a1.1, 2.2.4.22a1.10

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:	\( x^{8} + 16 t x^{7} + 8 x^{2} + 16 x + 16 t + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[24, 16, 8, 0]$

Invariants of the Galois closure

Galois degree:	$32$
Galois group:	$C_8.C_4$ (as 16T49)
Inertia group:	not computed
Wild inertia group:	not computed
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[3, 4, 5]$
Galois Swan slopes:	$[2,3,4]$
Galois mean slope:	$3.875$
Galois splitting model:	not computed