$p$-adic family 41.4.4.12a

• Label: 41.4.4.12a
• Base: 41.1.1.0a1.1
• Degree: \(16\)
• e: \(4\)
• f: \(4\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{4} + 41d_{0}$

Invariants

Residue field characteristic:	$41$
Degree:	$16$
Base field:	$\Q_{41}$
Ramification index $e$:	$4$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$12$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$4$ (complete)
Ambiguity:	$16$
Mass:	$1$
Absolute Mass:	$1/4$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_{16}$ (as 16T1)	$C_4^2$ (as 16T4)	$C_2\times C_8$ (as 16T5)	Total
hidden slopes	$[\ ]$	2	1	1	4

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
41.4.4.12a1.1	2	$( x^{4} + 23 x + 6 )^{4} + 41 x^{3}$	$C_{16}$ (as 16T1)	$16$	$16$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
41.4.4.12a1.2	1	$( x^{4} + 23 x + 6 )^{4} + 41 x^{2}$	$C_8\times C_2$ (as 16T5)	$16$	$16$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
41.4.4.12a1.3	2	$( x^{4} + 23 x + 6 )^{4} + 41 x$	$C_{16}$ (as 16T1)	$16$	$16$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
41.4.4.12a1.4	1	$( x^{4} + 23 x + 6 )^{4} + 41$	$C_4^2$ (as 16T4)	$16$	$16$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined

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