$p$-adic family 7.1.21.21a

• Label: 7.1.21.21a
• Base: 7.1.1.0a1.1
• Degree: \(21\)
• e: \(21\)
• f: \(1\)
• c: \(21\)

Defining polynomial

$x^{21} + 7 a_{1} x + 7 d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$21$
Base field:	$\Q_{7}$
Ramification index $e$:	$21$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$21$
Artin slopes:	$[\frac{19}{18}]$
Swan slopes:	$[\frac{1}{18}]$
Means:	$\langle\frac{1}{21}\rangle$
Rams:	$(\frac{1}{6})$
Field count:	$6$ (complete)
Ambiguity:	$3$
Mass:	$6$
Absolute Mass:	$6$

Diagrams

Varying

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

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
7.1.21.21a1.1		$x^{21} + 7 x + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 6$	undefined
7.1.21.21a1.2		$x^{21} + 21 x + 7$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 4$	undefined
7.1.21.21a1.3		$x^{21} + 7 x + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 3$	undefined
7.1.21.21a1.4		$x^{21} + 21 x + 14$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.21a1.5		$x^{21} + 7 x + 21$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 2$	undefined
7.1.21.21a1.6		$x^{21} + 21 x + 21$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z + 6$	undefined

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