$p$-adic field 2.1.16.56k1.61

• Label: 2.1.16.56k1.61
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(56\)
• Galois group: $C_2^4:D_4$ (as 16T392)

Defining polynomial

\(x^{16} + 8 x^{9} + 8 x^{4} + 8 x^{2} + 10\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$56$
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:	$[3, \frac{7}{2}, \frac{7}{2}, 4]$
Visible Swan slopes:	$[2,\frac{5}{2},\frac{5}{2},3]$
Means:	$\langle1, \frac{7}{4}, \frac{17}{8}, \frac{41}{16}\rangle$
Rams:	$(2, 3, 3, 7)$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.10a1.8, 2.1.4.11a1.5, 2.1.4.11a1.7, 2.1.8.24b1.7, 2.1.8.26c1.37, 2.1.8.26c1.39

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Canonical tower

Unramified subfield:	$\Q_{2}$
Relative Eisenstein polynomial:	\( x^{16} + 8 x^{9} + 8 x^{4} + 8 x^{2} + 10 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^6 + 1$,$z + 1$
Associated inertia:	$1$,$2$,$1$
Indices of inseparability:	$[41, 34, 32, 16, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_2^4:D_4$ (as 16T392)
Inertia group:	$C_2^4:C_4$ (as 16T79)
Wild inertia group:	$C_2^4:C_4$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},3]$
Galois mean slope:	$3.59375$
Galois splitting model:	$x^{16} + 8 x^{14} - 8 x^{13} - 4 x^{12} + 144 x^{11} - 136 x^{10} - 576 x^{9} + 2818 x^{8} - 3360 x^{7} + 2200 x^{6} + 920 x^{5} + 780 x^{4} - 1600 x^{3} + 5200 x^{2} - 4000 x + 4075$