$p$-adic field 2.2.8.36b8.39

• Label: 2.2.8.36b8.39
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(36\)
• Galois group: $D_4^2:C_8$ (as 16T941)

Defining polynomial

$( x^{2} + x + 1 )^{8} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 ( x^{2} + x + 1 )^{3} + 6$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$36$
Discriminant root field:	$\Q_{2}(\sqrt{5})$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_4$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 2, 3]$
Visible Swan slopes:	$[1,1,2]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{11}{8}\rangle$
Rams:	$(1, 1, 5)$
Jump set:	$[1, 2, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.4a2.2, 2.2.4.12a4.2

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:	\( x^{8} + 4 x^{7} + 2 t x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{3} + 10 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^6 + z^2 + t$,$t z + 1$
Associated inertia:	$2$,$1$
Indices of inseparability:	$[11, 6, 4, 0]$

Invariants of the Galois closure

Galois degree:	$512$
Galois group:	$D_4^2:C_8$ (as 16T941)
Inertia group:	not computed
Wild inertia group:	not computed
Galois unramified degree:	$8$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, 3, 3]$
Galois Swan slopes:	$[1,1,1,1,2,2]$
Galois mean slope:	$2.71875$
Galois splitting model:	not computed