$p$-adic family 131.1.4.3a

• Label: 131.1.4.3a
• Base: 131.1.1.0a1.1
• Degree: \(4\)
• e: \(4\)
• f: \(1\)
• c: \(3\)

Defining polynomial

$x^{4} + 131d_{0}$

Invariants

Residue field characteristic:	$131$
Degree:	$4$
Base field:	$\Q_{131}$
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$3$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[2]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$D_4$ (as 4T3)
hidden slopes	$[\ ]^{2}$	2

Fields

Showing all 2

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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
131.1.4.3a1.1	2	$x^{4} + 131$	$D_{4}$ (as 4T3)	$8$	$2$	$[\ ]_{4}^{2}$	$[\ ]_{4}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
131.1.4.3a1.2	2	$x^{4} + 262$	$D_{4}$ (as 4T3)	$8$	$2$	$[\ ]_{4}^{2}$	$[\ ]_{4}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^3 + 4 z^2 + 6 z + 4$	undefined

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