$p$-adic family 5.7.2.7a

• Label: 5.7.2.7a
• Base: 5.1.1.0a1.1
• Degree: \(14\)
• e: \(2\)
• f: \(7\)
• c: \(7\)

Defining polynomial over unramified subextension

$x^{2} + 5d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$14$
Base field:	$\Q_{5}$
Ramification index $e$:	$2$
Residue field degree $f$:	$7$
Discriminant exponent $c$:	$7$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$14$
Mass:	$1$
Absolute Mass:	$1/7$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[1]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$C_{14}$ (as 14T1)
hidden slopes	$[\ ]$	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
5.7.2.7a1.1	2	$( x^{7} + 3 x + 3 )^{2} + 5 x$	$C_{14}$ (as 14T1)	$14$	$14$	$[\ ]_{2}^{7}$	$[\ ]_{2}^{7}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z + 2$	undefined
5.7.2.7a1.2	2	$( x^{7} + 3 x + 3 )^{2} + 5$	$C_{14}$ (as 14T1)	$14$	$14$	$[\ ]_{2}^{7}$	$[\ ]_{2}^{7}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z + 2$	undefined

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