$p$-adic field 2.2.8.52c5.181

• Label: 2.2.8.52c5.181
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(52\)
• Galois group: $C_2^6.D_4$ (as 16T940)

Defining polynomial

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

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$52$
Discriminant root field:	$\Q_{2}$
Root number:	$-1$
$\Aut(K/\Q_{2})$:	$C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, \frac{7}{2}, 4]$
Visible Swan slopes:	$[2,\frac{5}{2},3]$
Means:	$\langle1, \frac{7}{4}, \frac{19}{8}\rangle$
Rams:	$(2, 3, 5)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.6a1.4, 2.2.4.20a2.6, 2.2.4.22a1.48, 2.2.4.22a1.78

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

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z^2 + t$,$t z + (t + 1)$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[19, 14, 8, 0]$

Invariants of the Galois closure

Galois degree:	$512$
Galois group:	$C_2^6.D_4$ (as 16T940)
Inertia group:	Intransitive group isomorphic to $C_2^3:C_4^2$
Wild inertia group:	$C_2^3:C_4^2$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, 3, \frac{7}{2}, \frac{7}{2}, 4]$
Galois Swan slopes:	$[1,1,2,2,\frac{5}{2},\frac{5}{2},3]$
Galois mean slope:	$3.640625$
Galois splitting model:	$x^{16} - 12 x^{14} + 98 x^{12} - 504 x^{10} + 2375 x^{8} - 5664 x^{6} + 16718 x^{4} - 10572 x^{2} + 961$