$p$-adic field 2.1.16.48o2.33

• Label: 2.1.16.48o2.33
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(48\)
• Galois group: $C_2\wr A_4$ (as 16T425)

Defining polynomial

\(x^{16} + 4 x^{10} + 2 x^{8} + 4 x^{4} + 8 x + 6\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$48$
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:	$[2, 3, 3, \frac{7}{2}]$
Visible Swan slopes:	$[1,2,2,\frac{5}{2}]$
Means:	$\langle\frac{1}{2}, \frac{5}{4}, \frac{13}{8}, \frac{33}{16}\rangle$
Rams:	$(1, 3, 3, 7)$
Jump set:	$[1, 2, 4, 8, 32]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 2.1.8.20d2.3

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:	\( x^{16} + 4 x^{10} + 2 x^{8} + 4 x^{4} + 8 x + 6 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^6 + z^2 + 1$,$z + 1$
Associated inertia:	$1$,$3$,$1$
Indices of inseparability:	$[33, 26, 20, 8, 0]$

Invariants of the Galois closure

Galois degree:	$192$
Galois group:	$C_2\wr A_4$ (as 16T425)
Inertia group:	$C_2\wr C_2^2$ (as 16T147)
Wild inertia group:	$C_2\wr C_2^2$
Galois unramified degree:	$3$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 3, 3, \frac{7}{2}]$
Galois Swan slopes:	$[1,1,1,2,2,\frac{5}{2}]$
Galois mean slope:	$3.09375$
Galois splitting model:	$x^{16} - 16 x^{14} + 88 x^{12} - 208 x^{10} + 178 x^{8} - 16 x^{6} - 104 x^{4} + 80 x^{2} + 1$