$p$-adic family 109.1.14.13a

• Label: 109.1.14.13a
• Base: 109.1.1.0a1.1
• Degree: \(14\)
• e: \(14\)
• f: \(1\)
• c: \(13\)

Defining polynomial

$x^{14} + 109d_{0}$

Invariants

Residue field characteristic:	$109$
Degree:	$14$
Base field:	$\Q_{109}$
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:	$[3]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$C_7:C_6$ (as 14T5)
hidden slopes	$[\ ]^{3}$	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
109.1.14.13a1.1	2	$x^{14} + 109$	$(C_7:C_3) \times C_2$ (as 14T5)	$42$	$2$	$[\ ]_{14}^{3}$	$[\ ]_{14}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[3]$	$z^{13} + 14 z^{12} + 91 z^{11} + 37 z^{10} + 20 z^9 + 40 z^8 + 60 z^7 + 53 z^6 + 60 z^5 + 40 z^4 + 20 z^3 + 37 z^2 + 91 z + 14$	undefined
109.1.14.13a1.2	2	$x^{14} + 654$	$(C_7:C_3) \times C_2$ (as 14T5)	$42$	$2$	$[\ ]_{14}^{3}$	$[\ ]_{14}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[3]$	$z^{13} + 14 z^{12} + 91 z^{11} + 37 z^{10} + 20 z^9 + 40 z^8 + 60 z^7 + 53 z^6 + 60 z^5 + 40 z^4 + 20 z^3 + 37 z^2 + 91 z + 14$	undefined

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