$p$-adic family 2.1.4.4a

• Label: 2.1.4.4a
• Base: 2.1.1.0a1.1
• Degree: \(4\)
• e: \(4\)
• f: \(1\)
• c: \(4\)

Defining polynomial

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

Invariants

Residue field characteristic:	$2$
Degree:	$4$
Base field:	$\Q_{2}$
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$4$
Artin slopes:	$[\frac{4}{3},\frac{4}{3}]$
Swan slopes:	$[\frac{1}{3},\frac{1}{3}]$
Means:	$\langle\frac{1}{6},\frac{1}{4}\rangle$
Rams:	$(\frac{1}{3},\frac{1}{3})$
Field count:	$1$ (complete)
Ambiguity:	$1$
Mass:	$1$
Absolute Mass:	$1$

Diagrams

Varying

Indices of inseparability:	$[1,1,0]$
Associated inertia:	$[1]$
Jump Set:	$[1,2,5]$

Galois groups and Hidden Artin slopes

		Galois groups
		$S_4$ (as 4T5)
hidden slopes	$[\ ]^{2}_{3}$	1

Fields

Showing all 1

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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
2.1.4.4a1.1	1	$x^{4} + 2 x + 2$	$S_4$ (as 4T5)	$24$	$1$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{2}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{2}$	$[\ ]^{2}_{3}$	$[\ ]^{2}_{3}$	$[1, 1, 0]$	$[1]$	$z + 1$	$[1, 2, 5]$

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