$p$-adic field 2.16.71.2615

• Label: 2.16.71.2615
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(71\)
• Galois group: $C_2^5.C_2\wr D_4$ (as 16T1641)

Defining polynomial

\(x^{16} + 20 x^{12} + 16 x^{11} + 58 x^{8} + 32 x^{6} + 40 x^{4} + 16 x^{2} + 38\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification exponent $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$71$
Discriminant root field:	$\Q_{2}(\sqrt{-2})$
Root number:	$i$
$\card{ \Aut(K/\Q_{ 2 }) }$:	$2$
This field is not Galois over $\Q_{2}.$
Visible slopes:	$[2, 7/2, 9/2, 11/2]$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 2.4.9.6, 2.8.27.123

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:	$\Q_{2}$
Relative Eisenstein polynomial:	\( x^{16} + 20 x^{12} + 16 x^{11} + 58 x^{8} + 32 x^{6} + 40 x^{4} + 16 x^{2} + 38 \)

Ramification polygon

Residual polynomials:	$z + 1$,$z^{2} + 1$,$z^{4} + 1$,$z^{8} + 1$
Associated inertia:	$1$,$1$,$1$,$1$
Indices of inseparability:	$[56, 40, 24, 8, 0]$

Invariants of the Galois closure

Galois group:	$C_2^5.C_2\wr D_4$ (as 16T1641)
Inertia group:	$C_2^7.D_8$ (as 16T1454)
Wild inertia group:	data not computed
Unramified degree:	$2$
Tame degree:	$1$
Wild slopes:	$[2, 2, 3, 7/2, 7/2, 15/4, 9/2, 9/2, 37/8, 5, 11/2]$
Galois mean slope:	$5231/1024$
Galois splitting model:	$x^{16} + 8 x^{14} - 44 x^{12} + 192 x^{10} - 450 x^{8} + 576 x^{6} - 408 x^{4} + 144 x^{2} - 18$