$p$-adic field 2.1.16.50h1.69

• Label: 2.1.16.50h1.69
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(50\)
• Galois group: $C_4^2:C_2^2$ (as 16T111)

Defining polynomial

\(x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 26\)

Invariants

Base field:	$\Q_{2}$
Degree $d$:	$16$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$50$
Discriminant root field:	$\Q_{2}$
Root number:	$-1$
$\Aut(K/\Q_{2})$:	$C_2\times C_4$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[2, 2, 3, 4]$
Visible Swan slopes:	$[1,1,2,3]$
Means:	$\langle\frac{1}{2}, \frac{3}{4}, \frac{11}{8}, \frac{35}{16}\rangle$
Rams:	$(1, 1, 5, 13)$
Jump set:	$[1, 3, 6, 32, 48]$
Roots of unity:	$4 = 2^{ 2 }$

Intermediate fields

$\Q_{2}(\sqrt{-1})$, $\Q_{2}(\sqrt{2\cdot 5})$, $\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.6a1.1, 2.1.4.8b1.2, 2.1.4.8b1.3, 2.1.8.18b1.2, 2.1.8.24c1.32, 2.1.8.24c1.29

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:	\( x^{16} + 8 x^{15} + 2 x^{12} + 8 x^{9} + 8 x^{7} + 4 x^{6} + 8 x^{3} + 26 \)

Ramification polygon

Residual polynomials:	$z^{12} + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$2$,$1$,$1$
Indices of inseparability:	$[35, 22, 12, 12, 0]$

Invariants of the Galois closure

Galois degree:	$64$
Galois group:	$C_4^2:C_2^2$ (as 16T111)
Inertia group:	$D_4:C_2$ (as 16T11)
Wild inertia group:	$D_4:C_2$
Galois unramified degree:	$4$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, 4]$
Galois Swan slopes:	$[1,1,2,3]$
Galois mean slope:	$3.125$
Galois splitting model:	$x^{16} - 8 x^{12} + 136 x^{8} + 480 x^{4} + 400$