$p$-adic family 7.1.7.12a

• Label: 7.1.7.12a
• Base: 7.1.1.0a1.1
• Degree: \(7\)
• e: \(7\)
• f: \(1\)
• c: \(12\)

Defining polynomial

$x^{7} + 7 a_{6} x^{6} + 49 c_{7} + 7$

Invariants

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

Diagrams

Varying

Indices of inseparability:	$[6,0]$
Associated inertia:	$[1]$ (show 7), $[2]$ (show 1), $[3]$ (show 2), $[6]$ (show 2)
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 7
		$C_7$ (as 7T1)
hidden slopes	$[\ ]$	7

		Galois groups of order 14
		$D_7$ (as 7T2)
hidden slopes	$[\ ]^{2}$	1

		Galois groups of order 21
		$C_7:C_3$ (as 7T3)
hidden slopes	$[\ ]^{3}$	2

		Galois groups of order 42
		$F_7$ (as 7T4)
hidden slopes	$[\ ]^{6}$	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
7.1.7.12a3.1	1	$x^{7} + 21 x^{6} + 7$	$C_7:C_3$ (as 7T3)	$21$	$1$	$[2]^{3}$	$[1]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[3]$	$z^6 + 3$	undefined
7.1.7.12a5.1	1	$x^{7} + 35 x^{6} + 7$	$C_7:C_3$ (as 7T3)	$21$	$1$	$[2]^{3}$	$[1]^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[3]$	$z^6 + 5$	undefined

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