$p$-adic family 2.1.8.31a1.166-2.1.0a

• Label: 2.1.8.31a1.166-2.1.0a
• Base: 2.1.8.31a1.166
• Degree: \(2\)
• e: \(1\)
• f: \(2\)
• c: \(0\)

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.8.31a1.166
Ramification index $e$:	$1$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$0$
Absolute Artin slopes:	$[3,4,5]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$1$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1/8$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group:	$C_2 \times (C_8:C_2)$
Hidden Artin slopes:	$[2]$
Indices of inseparability:	$[24,16,8,0]$
Associated inertia:	$[1,1,1]$
Jump Set:	$[1,3,7,15]$

Fields

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
2.2.8.62a1.9321	$( x^{2} + x + 1 )^{8} + 8 ( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{4} + 2$	$C_2 \times (C_8:C_2)$ (as 16T15)	$32$	$8$	$[2, 3, 4, 5]^{2}$	$[1,2,3,4]^{2}$	$[2]$	$[1]$	$[24, 16, 8, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15]$

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