$p$-adic family 5.1.14.13a

• Label: 5.1.14.13a
• Base: 5.1.1.0a1.1
• Degree: \(14\)
• e: \(14\)
• f: \(1\)
• c: \(13\)

Defining polynomial

$x^{14} + 5d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$14$
Base field:	$\Q_{5}$
Ramification index $e$:	$14$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$13$
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:	$[6]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$C_2\times F_7$ (as 14T7)
hidden slopes	$[\ ]^{6}$	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
5.1.14.13a1.1	2	$x^{14} + 5$	$F_7 \times C_2$ (as 14T7)	$84$	$2$	$[\ ]_{14}^{6}$	$[\ ]_{14}^{6}$	$[\ ]^{6}$	$[\ ]^{6}$	$[0]$	$[6]$	$z^{13} + 4 z^{12} + z^{11} + 4 z^{10} + z^9 + 2 z^8 + 3 z^7 + 2 z^6 + 3 z^5 + 2 z^4 + z^3 + 4 z^2 + z + 4$	undefined
5.1.14.13a1.2	2	$x^{14} + 10$	$F_7 \times C_2$ (as 14T7)	$84$	$2$	$[\ ]_{14}^{6}$	$[\ ]_{14}^{6}$	$[\ ]^{6}$	$[\ ]^{6}$	$[0]$	$[6]$	$z^{13} + 4 z^{12} + z^{11} + 4 z^{10} + z^9 + 2 z^8 + 3 z^7 + 2 z^6 + 3 z^5 + 2 z^4 + z^3 + 4 z^2 + z + 4$	undefined

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