$p$-adic field 2.2.8.28a2.1

• Label: 2.2.8.28a2.1
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(28\)
• Galois group: $C_2\wr C_6$ (as 16T719)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$28$
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:	$[2, 2, 2]$
Visible Swan slopes:	$[1,1,1]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{7}{8}\rangle$
Rams:	$(1, 1, 1)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.4a2.1, 2.2.4.12a5.1

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

Ramification polygon

Residual polynomials:	$z^7 + t$
Associated inertia:	$3$
Indices of inseparability:	$[7, 7, 7, 0]$

Invariants of the Galois closure

Galois degree:	$384$
Galois group:	$C_2\wr C_6$ (as 16T719)
Inertia group:	Intransitive group isomorphic to $C_2^6$
Wild inertia group:	$C_2^6$
Galois unramified degree:	$6$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, 2, 2]$
Galois Swan slopes:	$[1,1,1,1,1,1]$
Galois mean slope:	$1.96875$
Galois splitting model:	$x^{16} - 4 x^{15} + 12 x^{14} - 12 x^{13} + 20 x^{12} - 184 x^{11} + 116 x^{10} - 12 x^{9} - 108 x^{8} + 328 x^{7} - 240 x^{5} + 12 x^{4} + 68 x^{3} - 8 x - 1$