$p$-adic family 191.2.4.6a

• Label: 191.2.4.6a
• Base: 191.1.1.0a1.1
• Degree: \(8\)
• e: \(4\)
• f: \(2\)
• c: \(6\)

Defining polynomial over unramified subextension

$x^{4} + 191d_{0}$

Invariants

Residue field characteristic:	$191$
Degree:	$8$
Base field:	$\Q_{191}$
Ramification index $e$:	$4$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$6$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$8$
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 8
		$D_4$ (as 8T4)	$Q_8$ (as 8T5)	Total
hidden slopes	$[\ ]$	1	1	2

		Galois groups of order 16
		$\OD_{16}$ (as 8T7)
hidden slopes	$[\ ]^{2}$	1

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
191.2.4.6a1.1	1	$( x^{2} + 190 x + 19 )^{4} + 191 x$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
191.2.4.6a1.2	1	$( x^{2} + 190 x + 19 )^{4} + 191$	$D_4$ (as 8T4)	$8$	$8$	$[\ ]_{4}^{2}$	$[\ ]_{4}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
191.2.4.6a1.3	1	$( x^{2} + 190 x + 19 )^{4} + 191 x + 32852$	$Q_8$ (as 8T5)	$8$	$8$	$[\ ]_{4}^{2}$	$[\ ]_{4}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined

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