$p$-adic family 7.6.3.12a

• Label: 7.6.3.12a
• Base: 7.1.1.0a1.1
• Degree: \(18\)
• e: \(3\)
• f: \(6\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{3} + 7d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$18$
Base field:	$\Q_{7}$
Ramification index $e$:	$3$
Residue field degree $f$:	$6$
Discriminant exponent $c$:	$12$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$18$
Mass:	$1$
Absolute Mass:	$1/6$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_{18}$ (as 18T1)	$C_3\times C_6$ (as 18T2)	Total
hidden slopes	$[\ ]$	2	1	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
7.6.3.12a1.1	2	$( x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3 )^{3} + 7 x^{2}$	$C_{18}$ (as 18T1)	$18$	$18$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
7.6.3.12a1.2	2	$( x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3 )^{3} + 7 x$	$C_{18}$ (as 18T1)	$18$	$18$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
7.6.3.12a1.3	1	$( x^{6} + x^{4} + 5 x^{3} + 4 x^{2} + 6 x + 3 )^{3} + 7$	$C_6 \times C_3$ (as 18T2)	$18$	$18$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined

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