$p$-adic family 3.1.9.19a

• Label: 3.1.9.19a
• Base: 3.1.1.0a1.1
• Degree: \(9\)
• e: \(9\)
• f: \(1\)
• c: \(19\)

Defining polynomial

$x^{9} + 9 b_{12} x^{3} + 9 a_{11} x^{2} + 3$

Invariants

Residue field characteristic:	$3$
Degree:	$9$
Base field:	$\Q_{3}$
Ramification index $e$:	$9$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$19$
Artin slopes:	$[\frac{19}{8},\frac{19}{8}]$
Swan slopes:	$[\frac{11}{8},\frac{11}{8}]$
Means:	$\langle\frac{11}{12},\frac{11}{9}\rangle$
Rams:	$(\frac{11}{8},\frac{11}{8})$
Field count:	$6$ (complete)
Ambiguity:	$1$
Mass:	$6$
Absolute Mass:	$6$

Diagrams

Varying

Indices of inseparability:	$[11,9,0]$
Associated inertia:	$[1]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$F_9:C_2$ (as 9T19)
hidden slopes	$[\ ]^{2}_{8}$	6

Fields

Showing all 6

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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
3.1.9.19a1.1	6	$x^{9} + 9 x^{2} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{19}{8}, \frac{19}{8}]_{8}^{2}$	$[\frac{11}{8},\frac{11}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[11, 9, 0]$	$[1]$	$z + 1$	undefined
3.1.9.19a1.2	6	$x^{9} + 9 x^{3} + 9 x^{2} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{19}{8}, \frac{19}{8}]_{8}^{2}$	$[\frac{11}{8},\frac{11}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[11, 9, 0]$	$[1]$	$z + 1$	undefined
3.1.9.19a1.3	6	$x^{9} + 18 x^{3} + 9 x^{2} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{19}{8}, \frac{19}{8}]_{8}^{2}$	$[\frac{11}{8},\frac{11}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[11, 9, 0]$	$[1]$	$z + 1$	undefined
3.1.9.19a1.4	6	$x^{9} + 18 x^{2} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{19}{8}, \frac{19}{8}]_{8}^{2}$	$[\frac{11}{8},\frac{11}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[11, 9, 0]$	$[1]$	$z + 1$	undefined
3.1.9.19a1.5	6	$x^{9} + 9 x^{3} + 18 x^{2} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{19}{8}, \frac{19}{8}]_{8}^{2}$	$[\frac{11}{8},\frac{11}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[11, 9, 0]$	$[1]$	$z + 1$	undefined
3.1.9.19a1.6	6	$x^{9} + 18 x^{3} + 18 x^{2} + 3$	$(C_3^2:C_8):C_2$ (as 9T19)	$144$	$1$	$[\frac{19}{8}, \frac{19}{8}]_{8}^{2}$	$[\frac{11}{8},\frac{11}{8}]_{8}^{2}$	$[\ ]^{2}_{8}$	$[\ ]^{2}_{8}$	$[11, 9, 0]$	$[1]$	$z + 1$	undefined

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