$p$-adic family 41.2.7.12a

• Label: 41.2.7.12a
• Base: 41.1.1.0a1.1
• Degree: \(14\)
• e: \(7\)
• f: \(2\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{7} + 41d_{0}$

Invariants

Residue field characteristic:	$41$
Degree:	$14$
Base field:	$\Q_{41}$
Ramification index $e$:	$7$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$12$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$4$ (complete)
Ambiguity:	$14$
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 14
		$D_7$ (as 14T2)
hidden slopes	$[\ ]$	1

		Galois groups of order 98
		$C_7\times D_7$ (as 14T8)
hidden slopes	$[\ ]^{7}$	3

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
41.2.7.12a1.1	3	$( x^{2} + 38 x + 6 )^{7} + 41 x$	$C_7 \wr C_2$ (as 14T8)	$98$	$7$	$[\ ]_{7}^{14}$	$[\ ]_{7}^{14}$	$[\ ]^{7}$	$[\ ]^{7}$	$[0]$	$[1]$	$z^6 + 7 z^5 + 21 z^4 + 35 z^3 + 35 z^2 + 21 z + 7$	undefined
41.2.7.12a1.2	1	$( x^{2} + 38 x + 6 )^{7} + 41$	$D_{7}$ (as 14T2)	$14$	$14$	$[\ ]_{7}^{2}$	$[\ ]_{7}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^6 + 7 z^5 + 21 z^4 + 35 z^3 + 35 z^2 + 21 z + 7$	undefined
41.2.7.12a1.3	3	$( x^{2} + 38 x + 6 )^{7} + 123 x + 943$	$C_7 \wr C_2$ (as 14T8)	$98$	$7$	$[\ ]_{7}^{14}$	$[\ ]_{7}^{14}$	$[\ ]^{7}$	$[\ ]^{7}$	$[0]$	$[1]$	$z^6 + 7 z^5 + 21 z^4 + 35 z^3 + 35 z^2 + 21 z + 7$	undefined
41.2.7.12a1.4	3	$( x^{2} + 38 x + 6 )^{7} + 123 x + 1435$	$C_7 \wr C_2$ (as 14T8)	$98$	$7$	$[\ ]_{7}^{14}$	$[\ ]_{7}^{14}$	$[\ ]^{7}$	$[\ ]^{7}$	$[0]$	$[1]$	$z^6 + 7 z^5 + 21 z^4 + 35 z^3 + 35 z^2 + 21 z + 7$	undefined

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