$p$-adic field 2.1.16.60o1.94

• Label: 2.1.16.60o1.94
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(60\)
• Galois group: $C_2^4.D_4$ (as 16T247)

Defining polynomial

\(x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{6} + 16 x^{4} + 10\)

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:	$-1$
$\Aut(K/\Q_{2})$:	$C_2^2$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, \frac{7}{2}, 4, \frac{17}{4}]$
Visible Swan slopes:	$[2,\frac{5}{2},3,\frac{13}{4}]$
Means:	$\langle1, \frac{7}{4}, \frac{19}{8}, \frac{45}{16}\rangle$
Rams:	$(2, 3, 5, 7)$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{2\cdot 5})$, 2.1.4.10a1.3, 2.1.4.11a1.15, 2.1.4.11a1.20, 2.1.8.26c1.15, 2.1.8.28b1.54, 2.1.8.28b1.49

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:	\( x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{13} + 4 x^{12} + 8 x^{6} + 16 x^{4} + 10 \)

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z^2 + 1$,$z + 1$
Associated inertia:	$1$,$1$,$1$,$1$
Indices of inseparability:	$[45, 38, 28, 16, 0]$

Invariants of the Galois closure

Galois degree:	$128$
Galois group:	$C_2^4.D_4$ (as 16T247)
Inertia group:	$C_2^2.C_4^2$ (as 16T96)
Wild inertia group:	$C_2^2.C_4^2$
Galois unramified degree:	$2$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]$
Galois Swan slopes:	$[1,2,2,\frac{5}{2},3,\frac{13}{4}]$
Galois mean slope:	$3.875$
Galois splitting model:	$x^{16} + 24 x^{14} - 24 x^{13} + 32 x^{12} - 192 x^{11} - 392 x^{10} - 1320 x^{9} - 158 x^{8} + 1056 x^{7} + 9872 x^{6} + 30216 x^{5} + 42936 x^{4} + 15072 x^{3} + 2752 x^{2} + 2184 x + 1791$