$p$-adic field 2.1.16.42b1.79

• Label: 2.1.16.42b1.79
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(42\)
• Galois group: $C_2\wr C_2^3:C_3$ (as 16T1658)

Defining polynomial

\(x^{16} + 2 x^{14} + 4 x^{11} + 8 x^{7} + 8 x^{5} + 6\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$42$
Discriminant root field:	$\Q_{2}(\sqrt{5})$
Root number:	$-1$
$\Aut(K/\Q_{2})$:	$C_2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 2, 2, \frac{7}{2}]$
Visible Swan slopes:	$[1,1,1,\frac{5}{2}]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \frac{27}{16}\rangle$
Rams:	$(1, 1, 1, 13)$
Jump set:	$[1, 3, 7, 14, 32]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{-5})$, 2.1.4.6a2.1, 2.1.8.14a1.2

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:	\( x^{16} + 2 x^{14} + 4 x^{11} + 8 x^{7} + 8 x^{5} + 6 \)

Ramification polygon

Residual polynomials:	$z^{14} + 1$,$z + 1$
Associated inertia:	$3$,$1$
Indices of inseparability:	$[27, 14, 14, 14, 0]$

Invariants of the Galois closure

Galois degree:	$6144$
Galois group:	$C_2\wr C_2^3:C_3$ (as 16T1658)
Inertia group:	$C_2^7:C_2^3$ (as 16T1082)
Wild inertia group:	not computed
Galois unramified degree:	$6$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 2, 2, 2, 2, \frac{5}{2}, 3, 3, \frac{7}{2}]$
Galois Swan slopes:	$[1,1,1,1,1,1,\frac{3}{2},2,2,\frac{5}{2}]$
Galois mean slope:	$3.154296875$
Galois splitting model:	$x^{16} - 4 x^{14} + 28 x^{12} - 56 x^{10} + 170 x^{8} - 260 x^{6} + 196 x^{4} - 64 x^{2} + 13$