$p$-adic field 2.4.4.44a1.27

• Label: 2.4.4.44a1.27
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(4\)
• f: \(4\)
• c: \(44\)
• Galois group: $C_2^6:(C_2\times C_4)$ (as 16T815)

Defining polynomial

$( x^{4} + x + 1 )^{4} + 8 x^{3} ( x^{4} + x + 1 )^{3} + 8 x ( x^{4} + x + 1 ) + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$4$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$44$
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:	$[3, 4]$
Visible Swan slopes:	$[2,3]$
Means:	$\langle1, 2\rangle$
Rams:	$(2, 4)$
Jump set:	$[1, 3, 7]$
Roots of unity:	$30 = (2^{ 4 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, $\Q_{2}(\sqrt{-2})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.4.1.0a1.1, 2.2.2.6a1.1, 2.2.2.6a1.2, 2.4.2.12a1.1

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

Canonical tower

Unramified subfield:	2.4.1.0a1.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{4} + x + 1 \)
Relative Eisenstein polynomial:	\( x^{4} + \left(8 t^{3} + 8\right) x^{3} + 8 t x + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

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

Invariants of the Galois closure

Galois degree:	$512$
Galois group:	$C_2^6:(C_2\times C_4)$ (as 16T815)
Inertia group:	Intransitive group isomorphic to $C_2^3\wr C_2$
Wild inertia group:	$C_2^3\wr C_2$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]$
Galois Swan slopes:	$[1,1,1,2,\frac{5}{2},\frac{5}{2},3]$
Galois mean slope:	$3.609375$
Galois splitting model:	$x^{16} - 16 x^{14} + 74 x^{12} + 8 x^{10} - 416 x^{8} - 1664 x^{6} + 4736 x^{4} + 5632 x^{2} + 1936$