$p$-adic family 109.1.5.4a1.1-1.4.3a

• Label: 109.1.5.4a1.1-1.4.3a
• Base: 109.1.5.4a1.1
• Degree: \(4\)
• e: \(4\)
• f: \(1\)
• c: \(3\)

Defining polynomial

$x^{4} + d_{0} \pi$

Invariants

Residue field characteristic:	$109$
Degree:	$4$
Base field:	109.1.5.4a1.1
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$3$
Absolute Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$4$ (complete)
Ambiguity:	$4$
Mass:	$1$
Absolute Mass:	$1$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 109 }$ within this relative family, not the relative extension.

Galois group:	$C_4\times D_5$
Hidden Artin slopes:	$[\ ]^{2}$
Indices of inseparability:	$[0]$
Associated inertia:	$[2]$
Jump Set:	undefined

Fields

Showing all 4

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
109.1.20.19a1.1	$x^{20} + 109$	$C_4\times D_5$ (as 20T6)	$40$	$4$	$[\ ]_{20}^{2}$	$[\ ]_{20}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{19} + 20 z^{18} + 81 z^{17} + 50 z^{16} + 49 z^{15} + 26 z^{14} + 65 z^{13} + 21 z^{12} + 75 z^{11} + 100 z^{10} + z^9 + 100 z^8 + 75 z^7 + 21 z^6 + 65 z^5 + 26 z^4 + 49 z^3 + 50 z^2 + 81 z + 20$	undefined
109.1.20.19a1.2	$x^{20} + 654$	$C_4\times D_5$ (as 20T6)	$40$	$4$	$[\ ]_{20}^{2}$	$[\ ]_{20}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{19} + 20 z^{18} + 81 z^{17} + 50 z^{16} + 49 z^{15} + 26 z^{14} + 65 z^{13} + 21 z^{12} + 75 z^{11} + 100 z^{10} + z^9 + 100 z^8 + 75 z^7 + 21 z^6 + 65 z^5 + 26 z^4 + 49 z^3 + 50 z^2 + 81 z + 20$	undefined
109.1.20.19a1.3	$x^{20} + 3924$	$C_4\times D_5$ (as 20T6)	$40$	$4$	$[\ ]_{20}^{2}$	$[\ ]_{20}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{19} + 20 z^{18} + 81 z^{17} + 50 z^{16} + 49 z^{15} + 26 z^{14} + 65 z^{13} + 21 z^{12} + 75 z^{11} + 100 z^{10} + z^9 + 100 z^8 + 75 z^7 + 21 z^6 + 65 z^5 + 26 z^4 + 49 z^3 + 50 z^2 + 81 z + 20$	undefined
109.1.20.19a1.4	$x^{20} + 11663$	$C_4\times D_5$ (as 20T6)	$40$	$4$	$[\ ]_{20}^{2}$	$[\ ]_{20}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{19} + 20 z^{18} + 81 z^{17} + 50 z^{16} + 49 z^{15} + 26 z^{14} + 65 z^{13} + 21 z^{12} + 75 z^{11} + 100 z^{10} + z^9 + 100 z^8 + 75 z^7 + 21 z^6 + 65 z^5 + 26 z^4 + 49 z^3 + 50 z^2 + 81 z + 20$	undefined

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