$p$-adic family 2.1.16.24c

• Label: 2.1.16.24c
• Base: 2.1.1.0a1.1
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(24\)

Defining polynomial

$x^{16} + 2 c_{12} x^{12} + 2 b_{11} x^{11} + 2 a_{9} x^{9} + 2 a_{6} x^{6} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$16$
Base field:	$\Q_{2}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$24$
Artin slopes:	$[\frac{10}{7},\frac{10}{7},\frac{10}{7},\frac{7}{4}]$
Swan slopes:	$[\frac{3}{7},\frac{3}{7},\frac{3}{7},\frac{3}{4}]$
Means:	$\langle\frac{3}{14},\frac{9}{28},\frac{3}{8},\frac{9}{16}\rangle$
Rams:	$(\frac{3}{7},\frac{3}{7},\frac{3}{7},3)$
Field count:	$4$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

Indices of inseparability:	$[9,6,6,6,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[1,2,4,11,25]$

Fields

Showing all 4

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Label	Packet size	Polynomial	Galois group	Galois degree	$\#\Aut(K/\Q_p)$	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
2.1.16.24c1.1		$x^{16} + 2 x^{9} + 2 x^{6} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[9, 6, 6, 6, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 2, 4, 11, 25]$
2.1.16.24c1.2		$x^{16} + 2 x^{12} + 2 x^{9} + 2 x^{6} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[9, 6, 6, 6, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 2, 4, 11, 25]$
2.1.16.24c1.3		$x^{16} + 2 x^{11} + 2 x^{9} + 2 x^{6} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[9, 6, 6, 6, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 2, 4, 11, 25]$
2.1.16.24c1.4		$x^{16} + 2 x^{12} + 2 x^{11} + 2 x^{9} + 2 x^{6} + 2$	$C_2^7.F_8:C_6$ (as 16T1841)	$43008$	$2$	not computed	not computed	not computed	not computed	$[9, 6, 6, 6, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 2, 4, 11, 25]$

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