$p$-adic field 2.2.8.36b7.22

• Label: 2.2.8.36b7.22
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(36\)
• Galois group: $D_4.A_4$ (as 16T180)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 4 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 2 ( x^{2} + x + 1 )^{4} + 4 x ( x^{2} + x + 1 )^{3} + 10$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$36$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2$
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

$\Q_{2}(\sqrt{5})$, 2.1.4.6a2.1, 2.2.4.12a3.1

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

Ramification polygon

Residual polynomials:	$z^6 + z^2 + 1$,$z + t$
Associated inertia:	$3$,$1$
Indices of inseparability:	$[11, 6, 4, 0]$

Invariants of the Galois closure

Galois degree:	$96$
Galois group:	$D_4.A_4$ (as 16T180)
Inertia group:	Intransitive group isomorphic to $C_2\times Q_8$
Wild inertia group:	$C_2\times Q_8$
Galois unramified degree:	$6$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, 3]$
Galois Swan slopes:	$[1,1,2,2]$
Galois mean slope:	$2.625$
Galois splitting model:	$x^{16} - 4 x^{15} + 2 x^{14} - 20 x^{13} + 76 x^{12} - 12 x^{11} + 494 x^{10} + 124 x^{9} - 440 x^{8} - 7876 x^{7} - 9562 x^{6} + 2076 x^{5} + 31564 x^{4} + 49580 x^{3} + 40298 x^{2} + 17284 x + 2947$