$p$-adic field 2.2.8.54a2.6696

• Label: 2.2.8.54a2.6696
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(54\)
• Galois group: $(C_2\times C_4^3).D_4$ (as 16T1112)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 6 ( x^{2} + x + 1 )^{6} + \left(4 x + 6\right) ( x^{2} + x + 1 )^{5} + \left(2 x + 16\right) ( x^{2} + x + 1 )^{4} + \left(16 x + 8\right) ( x^{2} + x + 1 )^{3} + 8 x ( x^{2} + x + 1 )^{2} + 16 ( x^{2} + x + 1 ) + 6$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$54$
Discriminant root field:	$\Q_{2}(\sqrt{-5})$
Root number:	$-i$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, \frac{7}{2}, \frac{9}{2}]$
Visible Swan slopes:	$[1,\frac{5}{2},\frac{7}{2}]$
Means:	$\langle\frac{1}{2}, \frac{3}{2}, \frac{5}{2}\rangle$
Rams:	$(1, 4, 8)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.4a2.2, 2.2.4.18a2.17

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:	\( x^{8} + \left(8 t + 8\right) x^{7} + 4 t x^{6} + 8 t x^{5} + \left(2 t + 24\right) x^{4} + 8 t x^{2} + \left(16 t + 16\right) x + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^4 + t$,$t z^2 + t$,$t z + t$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[20, 12, 4, 0]$

Invariants of the Galois closure

Galois degree:	$1024$
Galois group:	$(C_2\times C_4^3).D_4$ (as 16T1112)
Inertia group:	Intransitive group isomorphic to $C_4^3.C_2^2$
Wild inertia group:	$C_4^3.C_2^2$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{9}{2}, \frac{9}{2}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{7}{2},\frac{7}{2}]$
Galois mean slope:	$4.2734375$
Galois splitting model:	$x^{16} + 14 x^{12} - 76 x^{8} + 100 x^{4} - 20$