$p$-adic field 2.1.16.64f1.130

• Label: 2.1.16.64f1.130
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(64\)
• Galois group: $C_2^6:(C_4\times S_3)$ (as 16T1300)

Defining polynomial

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

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$64$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_1$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, 4, \frac{53}{12}, \frac{53}{12}]$
Visible Swan slopes:	$[2,3,\frac{41}{12},\frac{41}{12}]$
Means:	$\langle1, 2, \frac{65}{24}, \frac{49}{16}\rangle$
Rams:	$(2, 4, \frac{17}{3}, \frac{17}{3})$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{2})$, 2.1.4.11a1.9

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

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[49, 48, 32, 16, 0]$

Invariants of the Galois closure

Galois degree:	$1536$
Galois group:	$C_2^6:(C_4\times S_3)$ (as 16T1300)
Inertia group:	$C_2^6:C_{12}$ (as 16T1041)
Wild inertia group:	$C_2^6:C_4$
Galois unramified degree:	$2$
Galois tame degree:	$3$
Galois Artin slopes:	$[\frac{8}{3}, \frac{8}{3}, 3, \frac{23}{6}, \frac{23}{6}, 4, \frac{53}{12}, \frac{53}{12}]$
Galois Swan slopes:	$[\frac{5}{3},\frac{5}{3},2,\frac{17}{6},\frac{17}{6},3,\frac{41}{12},\frac{41}{12}]$
Galois mean slope:	$4.252604166666667$
Galois splitting model:	$x^{16} - 32 x^{14} - 32 x^{13} + 432 x^{12} + 704 x^{11} - 336 x^{10} + 768 x^{9} + 4148 x^{8} + 1008 x^{7} - 3888 x^{6} + 1632 x^{5} + 5616 x^{4} + 3088 x^{3} + 11936 x^{2} + 21872 x + 13762$