$p$-adic family 3.1.9.13a

• Label: 3.1.9.13a
• Base: 3.1.1.0a1.1
• Degree: \(9\)
• e: \(9\)
• f: \(1\)
• c: \(13\)

Defining polynomial

$x^{9} + 3a_{5} x^{5} + 3$

Invariants

Residue field characteristic:	$3$
Degree:	$9$
Base field:	$\Q_{3}$
Ramification index $e$:	$9$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$13$
Artin slopes:	$[\frac{13}{8},\frac{13}{8}]$
Swan slopes:	$[\frac{5}{8},\frac{5}{8}]$
Means:	$\langle\frac{5}{12},\frac{5}{9}\rangle$
Rams:	$(\frac{5}{8},\frac{5}{8})$
Field count:	$2$ (complete)
Ambiguity:	$1$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$F_9:C_2$ (as 9T19)
hidden slopes	$[\ ]^{2}_{8}$	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
3.1.9.13a1.1	2	$x^{9} + 3 x^{5} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{13}{8}, \frac{13}{8}]_{8}^{2}$	$[\frac{5}{8},\frac{5}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[5, 5, 0]$	$[1]$	$z + 1$	undefined
3.1.9.13a1.2	2	$x^{9} + 6 x^{5} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{13}{8}, \frac{13}{8}]_{8}^{2}$	$[\frac{5}{8},\frac{5}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[5, 5, 0]$	$[1]$	$z + 1$	undefined

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