$p$-adic family 2.2.2.4a

• Label: 2.2.2.4a
• Base: 2.1.1.0a1.1
• Degree: \(4\)
• e: \(2\)
• f: \(2\)
• c: \(4\)

Defining polynomial over unramified subextension

$x^{2} + 2 a_{1} x + 4 c_{2} + 2$

Invariants

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

Diagrams

Varying

Indices of inseparability:	$[1,0]$
Associated inertia:	$[1]$
Jump Set:	$[1,2]$ (show 1), $[1,3]$ (show 2), $[1,4]$ (show 1)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 4
		$C_4$ (as 4T1)	$C_2^2$ (as 4T2)	Total
hidden slopes	$[\ ]$	1	1	2

		Galois groups of order 8
		$D_4$ (as 4T3)
hidden slopes	$[2]$	2

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
2.2.2.4a1.1	1	$( x^{2} + x + 1 )^{2} + 2 ( x^{2} + x + 1 ) + 2$	$C_2^2$ (as 4T2)	$4$	$4$	$[2]^{2}$	$[1]^{2}$	$[\ ]$	$[\ ]$	$[1, 0]$	$[1]$	$z + 1$	$[1, 2]$
2.2.2.4a1.2	1	$( x^{2} + x + 1 )^{2} + 2 ( x^{2} + x + 1 ) + 4 x + 2$	$C_4$ (as 4T1)	$4$	$4$	$[2]^{2}$	$[1]^{2}$	$[\ ]$	$[\ ]$	$[1, 0]$	$[1]$	$z + 1$	$[1, 4]$
2.2.2.4a2.1	2	$( x^{2} + x + 1 )^{2} + 2 x ( x^{2} + x + 1 ) + 2$	$D_{4}$ (as 4T3)	$8$	$2$	$[2, 2]^{2}$	$[1,1]^{2}$	$[2]$	$[1]$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.2.2.4a2.2	2	$( x^{2} + x + 1 )^{2} + 2 x ( x^{2} + x + 1 ) + 6$	$D_{4}$ (as 4T3)	$8$	$2$	$[2, 2]^{2}$	$[1,1]^{2}$	$[2]$	$[1]$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$

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