$p$-adic family 7.1.3.2a1.1-1.7.12a

• Label: 7.1.3.2a1.1-1.7.12a
• Base: 7.1.3.2a1.1
• Degree: \(7\)
• e: \(7\)
• f: \(1\)
• c: \(12\)

Defining polynomial

$x^{7} + a_{6} \pi x^{6} + c_{7} \pi^{2} + \pi$

Invariants

Residue field characteristic:	$7$
Degree:	$7$
Base field:	7.1.3.2a1.1
Ramification index $e$:	$7$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$12$
Absolute Artin slopes:	$[\frac{4}{3}]$
Swan slopes:	$[1]$
Means:	$\langle\frac{6}{7}\rangle$
Rams:	$(1)$
Field count:	$8$ (complete)
Ambiguity:	$7$
Mass:	$6$
Absolute Mass:	$2$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 7 }$ within this relative family, not the relative extension.

Indices of inseparability:	$[6,0]$
Associated inertia:	$[1,1]$ (show 3), $[1,2]$ (show 1), $[1,3]$ (show 2), $[1,6]$ (show 2)
Jump Set:	undefined

Fields

Showing all 8

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
7.1.21.26a1.1	$x^{21} + 7 x^{6} + 7$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^6 + 1$	undefined
7.1.21.26a2.1	$x^{21} + 14 x^{6} + 7$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 3]$	$z^{14} + 3 z^7 + 3,3 z^6 + 2$	undefined
7.1.21.26a3.1	$x^{21} + 21 x^{6} + 7$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 2]$	$z^{14} + 3 z^7 + 3,3 z^6 + 3$	undefined
7.1.21.26a4.1	$x^{21} + 28 x^{6} + 7$	$C_7:C_3$ (as 21T2)	$21$	$21$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z^6 + 4$	undefined
7.1.21.26a4.2	$x^{21} + 7 x^{7} + 28 x^{6} + 7$	$C_7:C_{21}$ (as 21T13)	$147$	$7$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z^6 + 4$	undefined
7.1.21.26a4.3	$x^{21} + 21 x^{7} + 28 x^{6} + 7$	$C_7:C_{21}$ (as 21T13)	$147$	$7$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 1]$	$z^{14} + 3 z^7 + 3,3 z^6 + 4$	undefined
7.1.21.26a5.1	$x^{21} + 35 x^{6} + 7$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 6]$	$z^{14} + 3 z^7 + 3,3 z^6 + 5$	undefined
7.1.21.26a6.1	$x^{21} + 42 x^{6} + 7$	$C_3\times F_7$ (as 21T9)	$126$	$3$	not computed	not computed	not computed	not computed	$[6, 0]$	$[1, 6]$	$z^{14} + 3 z^7 + 3,3 z^6 + 6$	undefined

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