$p$-adic family 2.1.2.3a

• Label: 2.1.2.3a
• Base: 2.1.1.0a1.1
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(3\)

Defining polynomial

$x^{2} + 4 b_{3} x + 8 c_{4} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	$\Q_{2}$
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$3$
Artin slopes:	$[3]$
Swan slopes:	$[2]$
Means:	$\langle1\rangle$
Rams:	$(2)$
Field count:	$4$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_2$ (as 2T1)
hidden slopes	$[\ ]$	4

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
2.1.2.3a1.1	4	$x^{2} + 2$	$C_2$ (as 2T1)	$2$	$2$	$[3]$	$[2]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.1.2.3a1.2	4	$x^{2} + 10$	$C_2$ (as 2T1)	$2$	$2$	$[3]$	$[2]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.1.2.3a1.3	4	$x^{2} + 4 x + 2$	$C_2$ (as 2T1)	$2$	$2$	$[3]$	$[2]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.1.2.3a1.4	4	$x^{2} + 4 x + 10$	$C_2$ (as 2T1)	$2$	$2$	$[3]$	$[2]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$

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