$p$-adic field 2.16.48.1

• Label: 2.16.48.1
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(48\)
• Galois group: $C_4\times C_2^2$ (as 16T2)

Defining polynomial

\(x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification exponent $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$48$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\card{ \Gal(K/\Q_{ 2 }) }$:	$16$
This field is Galois and abelian over $\Q_{2}.$
Visible slopes:	$[2, 3, 4]$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2\cdot 5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.4.8.2, 2.4.11.1, 2.4.6.1, 2.4.6.2, 2.4.4.1, 2.4.8.4, 2.4.8.1, 2.4.8.3, 2.4.11.2, 2.4.11.3, 2.4.11.4, 2.8.24.5, 2.8.16.6, 2.8.22.7, 2.8.22.2, 2.8.24.7, 2.8.24.10, 2.8.24.9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:	$\Q_{2}(\sqrt{5})$ $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{2} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{8} + 8 t x^{7} + \left(8 t + 8\right) x^{5} + \left(8 t + 10\right) x^{4} + \left(8 t + 4\right) x^{2} + 8 x + 16 t + 26 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

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

Invariants of the Galois closure

Galois group:	$C_2^2\times C_4$ (as 16T2)
Inertia group:	Intransitive group isomorphic to $C_2\times C_4$
Wild inertia group:	$C_2\times C_4$
Unramified degree:	$2$
Tame degree:	$1$
Wild slopes:	$[2, 3, 4]$
Galois mean slope:	$3$
Galois splitting model:	$x^{16} - x^{8} + 1$