$p$-adic field 2.1.16.60j1.110

• Label: 2.1.16.60j1.110
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(60\)
• Galois group: $C_2^6:(C_4\times S_3)$ (as 16T1300)

Defining polynomial

\(x^{16} + 8 x^{13} + 8 x^{12} + 8 x^{10} + 4 x^{8} + 26\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$60$
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:	$[3, 4, \frac{49}{12}, \frac{49}{12}]$
Visible Swan slopes:	$[2,3,\frac{37}{12},\frac{37}{12}]$
Means:	$\langle1, 2, \frac{61}{24}, \frac{45}{16}\rangle$
Rams:	$(2, 4, \frac{13}{3}, \frac{13}{3})$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.11a1.18

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:	\( x^{16} + 8 x^{13} + 8 x^{12} + 8 x^{10} + 4 x^{8} + 26 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$
Indices of inseparability:	$[45, 42, 32, 16, 0]$

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:	$[\frac{4}{3}, \frac{4}{3}, 3, \frac{19}{6}, \frac{19}{6}, 4, \frac{49}{12}, \frac{49}{12}]$
Galois Swan slopes:	$[\frac{1}{3},\frac{1}{3},2,\frac{13}{6},\frac{13}{6},3,\frac{37}{12},\frac{37}{12}]$
Galois mean slope:	$3.9244791666666665$
Galois splitting model:	$x^{16} - 8 x^{15} + 40 x^{14} - 80 x^{13} + 400 x^{12} - 704 x^{11} + 4432 x^{10} + 640 x^{9} + 72280 x^{8} + 151200 x^{7} + 586272 x^{6} + 1545504 x^{5} + 3048960 x^{4} + 5760000 x^{3} + 9279360 x^{2} + 12296448 x + 10236456$