$p$-adic family 2.1.18.22a

• Label: 2.1.18.22a
• Base: 2.1.1.0a1.1
• Degree: \(18\)
• e: \(18\)
• f: \(1\)
• c: \(22\)

Defining polynomial

$x^{18} + 2 c_{10} x^{10} + 2 b_{9} x^{9} + 2 b_{7} x^{7} + 2 a_{5} x^{5} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$18$
Base field:	$\Q_{2}$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$22$
Artin slopes:	$[\frac{14}{9}]$
Swan slopes:	$[\frac{5}{9}]$
Means:	$\langle\frac{5}{18}\rangle$
Rams:	$(5)$
Field count:	$8$ (complete)
Ambiguity:	$2$
Mass:	$4$
Absolute Mass:	$4$

Diagrams

Varying

Indices of inseparability:	$[5,0]$
Associated inertia:	$[6,1]$
Jump Set:	$[9,23]$

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 3456
		$C_2^6:C_9:C_6$ (as 18T433)	$C_2^6:C_9:C_6$ (as 18T434)	Total
hidden slopes	$[\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	2	2	4

		Galois groups of order 13824
		$C_2^2:A_4^2.S_4$ (as 18T588)	$C_2^2:A_4^2.S_4$ (as 18T592)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	2	2	4

Fields

Showing all 8

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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
2.1.18.22a1.1	2	$x^{18} + 2 x^{5} + 2$	$C_2^6:C_9:C_6$ (as 18T433)	$3456$	$2$	$[\frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.2	2	$x^{18} + 2 x^{10} + 2 x^{5} + 2$	$C_2^6:C_9:C_6$ (as 18T434)	$3456$	$2$	$[\frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.3	2	$x^{18} + 2 x^{9} + 2 x^{5} + 2$	$C_2^6:C_9:C_6$ (as 18T433)	$3456$	$2$	$[\frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.4	2	$x^{18} + 2 x^{10} + 2 x^{9} + 2 x^{5} + 2$	$C_2^6:C_9:C_6$ (as 18T434)	$3456$	$2$	$[\frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.5	2	$x^{18} + 2 x^{7} + 2 x^{5} + 2$	$C_2^2:A_4^2.S_4$ (as 18T588)	$13824$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.6	2	$x^{18} + 2 x^{10} + 2 x^{7} + 2 x^{5} + 2$	$C_2^2:A_4^2.S_4$ (as 18T592)	$13824$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.7	2	$x^{18} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2$	$C_2^2:A_4^2.S_4$ (as 18T588)	$13824$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$
2.1.18.22a1.8	2	$x^{18} + 2 x^{10} + 2 x^{9} + 2 x^{7} + 2 x^{5} + 2$	$C_2^2:A_4^2.S_4$ (as 18T592)	$13824$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}, \frac{14}{9}]_{9}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]_{9}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9},\frac{14}{9}]^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9},\frac{5}{9}]^{6}$	$[5, 0]$	$[6, 1]$	$z^{16} + z^{14} + 1,z + 1$	$[9, 23]$

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