$p$-adic family 7.1.9.8a

• Label: 7.1.9.8a
• Base: 7.1.1.0a1.1
• Degree: \(9\)
• e: \(9\)
• f: \(1\)
• c: \(8\)

Defining polynomial

$x^{9} + 7d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$9$
Base field:	$\Q_{7}$
Ramification index $e$:	$9$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$8$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$3$
Mass:	$1$
Absolute Mass:	$1$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[3]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$C_9:C_3$ (as 9T6)
hidden slopes	$[\ ]^{3}$	3

Fields

Showing all 3

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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
7.1.9.8a1.1	3	$x^{9} + 7$	$C_9:C_3$ (as 9T6)	$27$	$3$	$[\ ]_{9}^{3}$	$[\ ]_{9}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[3]$	$z^8 + 2 z^7 + z^6 + z + 2$	undefined
7.1.9.8a1.2	3	$x^{9} + 14$	$C_9:C_3$ (as 9T6)	$27$	$3$	$[\ ]_{9}^{3}$	$[\ ]_{9}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[3]$	$z^8 + 2 z^7 + z^6 + z + 2$	undefined
7.1.9.8a1.3	3	$x^{9} + 21$	$C_9:C_3$ (as 9T6)	$27$	$3$	$[\ ]_{9}^{3}$	$[\ ]_{9}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[3]$	$z^8 + 2 z^7 + z^6 + z + 2$	undefined

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