$p$-adic field 2.4.4.44a1.733

• Label: 2.4.4.44a1.733
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(4\)
• f: \(4\)
• c: \(44\)
• Galois group: $(C_4\times C_8).C_2^3$ (as 16T577)

Defining polynomial

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

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$4$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$44$
Discriminant root field:	$\Q_{2}(\sqrt{5})$
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

$\Q_{2}(\sqrt{5})$, 2.4.1.0a1.1, 2.4.2.12a1.6

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

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:	$256$
Galois group:	$(C_4\times C_8).C_2^3$ (as 16T577)
Inertia group:	Intransitive group isomorphic to $C_4^2:C_2^2$
Wild inertia group:	$C_4^2:C_2^2$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 3, 3, 4]$
Galois Swan slopes:	$[1,1,1,2,2,3]$
Galois mean slope:	$3.34375$
Galois splitting model:	$x^{16} - 20 x^{12} + 200 x^{8} - 1000 x^{4} + 2000$