$p$-adic family 5.2.6.10a

• Label: 5.2.6.10a
• Base: 5.1.1.0a1.1
• Degree: \(12\)
• e: \(6\)
• f: \(2\)
• c: \(10\)

Defining polynomial over unramified subextension

$x^{6} + 5d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$12$
Base field:	$\Q_{5}$
Ramification index $e$:	$6$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$10$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$4$ (complete)
Ambiguity:	$12$
Mass:	$1$
Absolute Mass:	$1/2$

Varying

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

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 12
		$D_6$ (as 12T3)	$C_3:C_4$ (as 12T5)	Total
hidden slopes	$[\ ]$	1	1	2

		Galois groups of order 36
		$C_6\times S_3$ (as 12T18)	$C_3:C_{12}$ (as 12T19)	Total
hidden slopes	$[\ ]^{3}$	1	1	2

Fields

Showing all 4

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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
5.2.6.10a1.1	1	$( x^{2} + 4 x + 2 )^{6} + 5 x$	$C_3\times (C_3 : C_4)$ (as 12T19)	$36$	$6$	$[\ ]_{6}^{6}$	$[\ ]_{6}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[1]$	$z^5 + z^4 + 1$	undefined
5.2.6.10a1.2	1	$( x^{2} + 4 x + 2 )^{6} + 5$	$D_6$ (as 12T3)	$12$	$12$	$[\ ]_{6}^{2}$	$[\ ]_{6}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + z^4 + 1$	undefined
5.2.6.10a1.3	1	$( x^{2} + 4 x + 2 )^{6} + 5 x + 15$	$C_6\times S_3$ (as 12T18)	$36$	$6$	$[\ ]_{6}^{6}$	$[\ ]_{6}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[1]$	$z^5 + z^4 + 1$	undefined
5.2.6.10a1.4	1	$( x^{2} + 4 x + 2 )^{6} + 20 x + 15$	$C_3 : C_4$ (as 12T5)	$12$	$12$	$[\ ]_{6}^{2}$	$[\ ]_{6}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^5 + z^4 + 1$	undefined

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