$p$-adic field 2.2.8.56a3.73

• Label: 2.2.8.56a3.73
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(56\)
• Galois group: $C_8:C_8:C_4$ (as 16T690)

Defining polynomial

$( x^{2} + x + 1 )^{8} + \left(8 x + 12\right) ( x^{2} + x + 1 )^{7} + 4 x ( x^{2} + x + 1 )^{6} + 8 ( x^{2} + x + 1 )^{5} + 8 ( x^{2} + x + 1 )^{4} + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$56$
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:	$[3, \frac{7}{2}, \frac{9}{2}]$
Visible Swan slopes:	$[2,\frac{5}{2},\frac{7}{2}]$
Means:	$\langle1, \frac{7}{4}, \frac{21}{8}\rangle$
Rams:	$(2, 3, 7)$
Jump set:	$[1, 3, 7, 15]$
Roots of unity:	$6 = (2^{ 2 } - 1) \cdot 2$

Intermediate fields

$\Q_{2}(\sqrt{5})$, 2.2.2.6a1.3, 2.2.4.20a2.2

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:	\( x^{8} + \left(8 t + 8\right) x^{7} + 4 t x^{6} + 8 x^{5} + 16 x^{4} + 16 t x^{3} + 8 t x^{2} + 16 x + 2 \) $\ \in\Q_{2}(t)[x]$

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z^2 + t$,$t z + 1$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[21, 14, 8, 0]$

Invariants of the Galois closure

Galois degree:	$256$
Galois group:	$C_8:C_8:C_4$ (as 16T690)
Inertia group:	Intransitive group isomorphic to $C_8:D_8$
Wild inertia group:	$C_8:D_8$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, \frac{9}{2}, \frac{9}{2}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},\frac{7}{2},\frac{7}{2}]$
Galois mean slope:	$4.171875$
Galois splitting model:	$x^{16} - 16 x^{14} + 100 x^{12} - 280 x^{10} + 260 x^{8} + 144 x^{6} - 64 x^{4} - 40 x^{2} - 5$