$p$-adic field 2.1.16.62c1.217

• Label: 2.1.16.62c1.217
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(62\)
• Galois group: $C_2^6:D_4$ (as 16T969)

Defining polynomial

\(x^{16} + 8 x^{15} + 16 x^{11} + 8 x^{8} + 16 x^{7} + 16 x^{5} + 8 x^{4} + 8 x^{2} + 2\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$62$
Discriminant root field:	$\Q_{2}$
Root number:	$1$
$\Aut(K/\Q_{2})$:	$C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, \frac{7}{2}, \frac{7}{2}, \frac{19}{4}]$
Visible Swan slopes:	$[2,\frac{5}{2},\frac{5}{2},\frac{15}{4}]$
Means:	$\langle1, \frac{7}{4}, \frac{17}{8}, \frac{47}{16}\rangle$
Rams:	$(2, 3, 3, 13)$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{-2})$, 2.1.4.10a1.5, 2.1.8.24b1.4, 2.1.8.29a1.80, 2.1.8.29a1.75

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:	\( x^{16} + 8 x^{15} + 16 x^{11} + 8 x^{8} + 16 x^{7} + 16 x^{5} + 8 x^{4} + 8 x^{2} + 2 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^6 + 1$,$z + 1$
Associated inertia:	$1$,$2$,$1$
Indices of inseparability:	$[47, 34, 32, 16, 0]$

Invariants of the Galois closure

Galois degree:	$512$
Galois group:	$C_2^6:D_4$ (as 16T969)
Inertia group:	$C_2^6:C_4$ (as 16T648)
Wild inertia group:	$C_2^6:C_4$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{19}{4}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{15}{4}]$
Galois mean slope:	$4.3359375$
Galois splitting model:	$x^{16} - 32 x^{14} + 260 x^{12} + 768 x^{10} - 9526 x^{8} - 72288 x^{6} + 656780 x^{4} - 1060928 x^{2} + 841$