$p$-adic field 2.1.16.54o1.191

• Label: 2.1.16.54o1.191
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(54\)
• Galois group: $C_4^2.D_4$ (as 16T344)

Defining polynomial

\(x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 10\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$54$
Discriminant root field:	$\Q_{2}$
Root number:	$-1$
$\Aut(K/\Q_{2})$:	$C_4$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 3, \frac{7}{2}, 4]$
Visible Swan slopes:	$[1,2,\frac{5}{2},3]$
Means:	$\langle\frac{1}{2}, \frac{5}{4}, \frac{15}{8}, \frac{39}{16}\rangle$
Rams:	$(1, 3, 5, 9)$
Jump set:	$[1, 7, 14, 32, 48]$
Roots of unity:	$4 = 2^{ 2 }$

Intermediate fields

$\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2\cdot 5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.8b1.2, 2.1.4.9a1.2, 2.1.4.9a1.1, 2.1.8.22d1.16

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:	\( x^{16} + 8 x^{15} + 4 x^{14} + 4 x^{12} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 10 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$,$1$
Indices of inseparability:	$[39, 30, 20, 8, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_4^2.D_4$ (as 16T344)
Inertia group:	$C_2^3.D_4$ (as 16T177)
Wild inertia group:	$C_2^3.D_4$
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:	$x^{16} - 40 x^{12} + 1450 x^{8} - 9000 x^{4} + 15625$