$p$-adic family 2.1.14.20a

• Label: 2.1.14.20a
• Base: 2.1.1.0a1.1
• Degree: \(14\)
• e: \(14\)
• f: \(1\)
• c: \(20\)

Defining polynomial

$x^{14} + 2 b_{13} x^{13} + 2 b_{11} x^{11} + 2 b_{9} x^{9} + 2 a_{7} x^{7} + 4 c_{14} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$14$
Base field:	$\Q_{2}$
Ramification index $e$:	$14$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$20$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(7)$
Field count:	$16$ (complete)
Ambiguity:	$2$
Mass:	$8$
Absolute Mass:	$8$

Diagrams

Varying

Indices of inseparability:	$[7,0]$
Associated inertia:	$[3,1]$
Jump Set:	$[7,14]$ (show 1), $[7,23]$ (show 8), $[7,25]$ (show 4), $[7,27]$ (show 2), $[7,28]$ (show 1)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 42
		$C_7:C_6$ (as 14T5)
hidden slopes	$[\ ]^{3}$	2

		Galois groups of order 336
		$F_8:C_6$ (as 14T18)
hidden slopes	$[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$	2
		$[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$	4
	$[\frac{8}{7},\frac{8}{7},\frac{8}{7}]^{3}$		2
		Total	8

		Galois groups of order 2688
		$C_2\wr C_7:C_3$ (as 14T44)
hidden slopes	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$	2
		$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$	4
	Total		6

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
2.1.14.20a1.5	2	$x^{14} + 2 x^{11} + 2 x^{7} + 2$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$	$[\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$	$[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$	$[\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 25]$
2.1.14.20a1.6	2	$x^{14} + 2 x^{11} + 2 x^{7} + 6$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$	$[\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$	$[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$	$[\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 25]$
2.1.14.20a1.7	2	$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 2$	$C_2\wr C_7:C_3$ (as 14T44)	$2688$	$2$	$[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 25]$
2.1.14.20a1.8	2	$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{7} + 6$	$C_2\wr C_7:C_3$ (as 14T44)	$2688$	$2$	$[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}, 2]_{7}^{3}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7},1]_{7}^{3}$	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 25]$

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