$p$-adic field 2.2.8.48c3.443

• Label: 2.2.8.48c3.443
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(48\)
• Galois group: $C_4^2.(C_2\times C_4)$ (as 16T362)

Defining polynomial

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

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$48$
Discriminant root field:	$\Q_{2}$
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:	$[1, 2, 4, 16]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.4a1.2, 2.2.4.16b2.9

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

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_4^2.(C_2\times C_4)$ (as 16T362)
Inertia group:	Intransitive group isomorphic to $C_8:C_4$
Wild inertia group:	$C_8:C_4$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 3, 3, 4, 4]$
Galois Swan slopes:	$[1,2,2,3,3]$
Galois mean slope:	$3.625$
Galois splitting model:	$x^{16} + 10 x^{12} + 10 x^{8} - 50 x^{4} + 25$