$p$-adic field 2.1.16.65c1.7309

• Label: 2.1.16.65c1.7309
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(65\)
• Galois group: $C_2\wr Q_8$ (as 16T1484)

Defining polynomial

\(x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 14\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$65$
Discriminant root field:	$\Q_{2}(\sqrt{-2})$
Root number:	$-i$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 3, 4, \frac{41}{8}]$
Visible Swan slopes:	$[1,2,3,\frac{33}{8}]$
Means:	$\langle\frac{1}{2}, \frac{5}{4}, \frac{17}{8}, \frac{25}{8}\rangle$
Rams:	$(1, 3, 7, 16)$
Jump set:	$[1, 2, 4, 8, 32]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, $\Q_{2}(\sqrt{2})$, 2.1.4.8b1.6, 2.1.8.24c1.56

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:	\( x^{16} + 16 x^{15} + 8 x^{14} + 16 x^{13} + 4 x^{12} + 16 x^{9} + 2 x^{8} + 4 x^{4} + 8 x^{2} + 14 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$,$1$
Indices of inseparability:	$[50, 34, 20, 8, 0]$

Invariants of the Galois closure

Galois degree:	$2048$
Galois group:	$C_2\wr Q_8$ (as 16T1484)
Inertia group:	$C_2^7.Q_8$ (as 16T1210)
Wild inertia group:	not computed
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 3, 3, \frac{7}{2}, 4, 4, \frac{17}{4}, \frac{19}{4}, 5, \frac{41}{8}]$
Galois Swan slopes:	$[1,2,2,\frac{5}{2},3,3,\frac{13}{4},\frac{15}{4},4,\frac{33}{8}]$
Galois mean slope:	$4.90625$
Galois splitting model:	$x^{16} - 60 x^{12} + 120 x^{10} + 594 x^{8} - 1296 x^{6} - 1476 x^{4} + 2664 x^{2} - 18$