$p$-adic field 2.1.16.62g2.47

• Label: 2.1.16.62g2.47
• Base: \(\Q_{2}\)
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(62\)
• Galois group: $C_2^6:(C_2\times A_4)$ (as 16T1299)

Defining polynomial

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

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:	$1$
$\Aut(K/\Q_{2})$:	$C_1$
This field is not Galois over $\Q_{2}.$
Visible Artin slopes:	$[3, 4, \frac{17}{4}, \frac{17}{4}]$
Visible Swan slopes:	$[2,3,\frac{13}{4},\frac{13}{4}]$
Means:	$\langle1, 2, \frac{21}{8}, \frac{47}{16}\rangle$
Rams:	$(2, 4, 5, 5)$
Jump set:	$[1, 3, 7, 15, 31]$
Roots of unity:	$2$

Intermediate fields

$\Q_{2}(\sqrt{-2\cdot 5})$, 2.1.4.11a1.8

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

Ramification polygon

Residual polynomials:	$z^8 + 1$,$z^4 + 1$,$z^3 + z + 1$
Associated inertia:	$1$,$1$,$3$
Indices of inseparability:	$[47, 42, 32, 16, 0]$

Invariants of the Galois closure

Galois degree:	$1536$
Galois group:	$C_2^6:(C_2\times A_4)$ (as 16T1299)
Inertia group:	$C_2^6:C_4$ (as 16T581)
Wild inertia group:	$C_2^6:C_4$
Galois unramified degree:	$6$
Galois tame degree:	$1$
Galois Artin slopes:	$[2, 2, 3, \frac{7}{2}, \frac{7}{2}, 4, \frac{17}{4}, \frac{17}{4}]$
Galois Swan slopes:	$[1,1,2,\frac{5}{2},\frac{5}{2},3,\frac{13}{4},\frac{13}{4}]$
Galois mean slope:	$4.0859375$
Galois splitting model:	$x^{16} - 48 x^{13} + 252 x^{12} - 1008 x^{11} + 3168 x^{10} - 10656 x^{9} + 30258 x^{8} - 64608 x^{7} + 108576 x^{6} - 132624 x^{5} + 140916 x^{4} - 121680 x^{3} + 105408 x^{2} - 70848 x + 34569$