$p$-adic field 2.2.8.56b2.2028

• Label: 2.2.8.56b2.2028
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(8\)
• f: \(2\)
• c: \(56\)
• Galois group: $C_2^6:D_4$ (as 16T956)

Defining polynomial

$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{7} + 12 ( x^{2} + x + 1 )^{6} + 12 ( x^{2} + x + 1 )^{5} + \left(4 x + 4\right) ( x^{2} + x + 1 )^{4} + 8 ( x^{2} + x + 1 )^{3} + \left(8 x + 24\right) ( x^{2} + x + 1 )^{2} + 24 ( x^{2} + x + 1 ) + 8 x + 2$

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$8$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$56$
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:	$[3, 4, \frac{17}{4}]$
Visible Swan slopes:	$[2,3,\frac{13}{4}]$
Means:	$\langle1, 2, \frac{21}{8}\rangle$
Rams:	$(2, 4, 5)$
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, 2.2.4.22a1.41

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

Ramification polygon

Residual polynomials:	$z^4 + 1$,$z^2 + 1$,$z + t$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[21, 16, 8, 0]$

Invariants of the Galois closure

Galois degree:	$512$
Galois group:	$C_2^6:D_4$ (as 16T956)
Inertia group:	Intransitive group isomorphic to $C_2^6:C_4$
Wild inertia group:	$C_2^6:C_4$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]$
Galois mean slope:	$4.0859375$
Galois splitting model:	$x^{16} + 32 x^{14} - 40 x^{13} + 452 x^{12} - 816 x^{11} + 3872 x^{10} - 4920 x^{9} + 13888 x^{8} - 16416 x^{7} - 50312 x^{6} + 33064 x^{5} + 31008 x^{4} + 3856 x^{3} + 95224 x^{2} + 108280 x + 32119$