$p$-adic family 2.2.4.22a1.81-2.1.0a

• Label: 2.2.4.22a1.81-2.1.0a
• Base: 2.2.4.22a1.81
• Degree: \(2\)
• e: \(1\)
• f: \(2\)
• c: \(0\)

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.2.4.22a1.81
Ramification index $e$:	$1$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$0$
Absolute Artin slopes:	$[3,4]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$1$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1/4$

Varying

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

Galois group:	$C_2^5:C_4$
Hidden Artin slopes:	$[2,2,\frac{7}{2}]$
Indices of inseparability:	$[8,4,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[1,3,7]$

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
2.4.4.44a1.1943	$( x^{4} + x + 1 )^{4} + \left(8 x + 12\right) ( x^{4} + x + 1 )^{3} + \left(4 x^{2} + 4 x + 4\right) ( x^{4} + x + 1 )^{2} + 2$	$C_2^5:C_4$ (as 16T227)	$128$	$4$	$[2, 2, 3, \frac{7}{2}, 4]^{4}$	$[1,1,2,\frac{5}{2},3]^{4}$	$[2,2,\frac{7}{2}]$	$[1,1,\frac{5}{2}]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$

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