$p$-adic family 2.3.6.18a

• Label: 2.3.6.18a
• Base: 2.1.1.0a1.1
• Degree: \(18\)
• e: \(6\)
• f: \(3\)
• c: \(18\)

Defining polynomial over unramified subextension

$x^{6} + 2 c_{2} x^{2} + 2 a_{1} x + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$18$
Base field:	$\Q_{2}$
Ramification index $e$:	$6$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$18$
Artin slopes:	$[\frac{4}{3}]$
Swan slopes:	$[\frac{1}{3}]$
Means:	$\langle\frac{1}{6}\rangle$
Rams:	$(1)$
Field count:	$6$ (complete)
Ambiguity:	$6$
Mass:	$7$
Absolute Mass:	$7/3$

Diagrams

Varying

Indices of inseparability:	$[1,0]$
Associated inertia:	$[2,1]$
Jump Set:	$[3,7]$

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 72
		$C_3\times S_4$ (as 18T30)	$C_3\times S_4$ (as 18T33)	Total
hidden slopes	$[\frac{4}{3}]^{2}$	1	1	2

		Galois groups of order 288
		$A_4\wr C_2$ (as 18T112)	$A_4\wr C_2$ (as 18T113)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	1	1	2

		Galois groups of order 1152
		$C_2^4:(C_3\times S_4)$ (as 18T269)	$C_2^4:(C_3\times S_4)$ (as 18T270)	Total
hidden slopes	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	1	1	2

Fields

Showing all 6

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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
2.3.6.18a1.1	1	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 ) + 2$	$C_3\times S_4$ (as 18T30)	$72$	$6$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{6}$	$[\frac{4}{3}]^{2}$	$[\frac{1}{3}]^{2}$	$[1, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + 1)$	$[3, 7]$
2.3.6.18a1.2	1	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{2} + 2 ( x^{3} + x + 1 ) + 2$	$C_3\times S_4$ (as 18T33)	$72$	$6$	$[\frac{4}{3}, \frac{4}{3}]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3}]_{3}^{6}$	$[\frac{4}{3}]^{2}$	$[\frac{1}{3}]^{2}$	$[1, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + (t^2 + 1)$	$[3, 7]$
2.3.6.18a2.1	1	$( x^{3} + x + 1 )^{6} + 2 x ( x^{3} + x + 1 ) + 2$	$C_2^4:(C_3\times S_4)$ (as 18T269)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[1, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 7]$
2.3.6.18a2.2	1	$( x^{3} + x + 1 )^{6} + 2 ( x^{3} + x + 1 )^{2} + 2 x ( x^{3} + x + 1 ) + 2$	$C_2^4:(C_3\times S_4)$ (as 18T270)	$1152$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[1, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 7]$
2.3.6.18a3.1	1	$( x^{3} + x + 1 )^{6} + \left(2 x + 2\right) ( x^{3} + x + 1 ) + 2$	$A_4\wr C_2$ (as 18T112)	$288$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[1, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + t^2$	$[3, 7]$
2.3.6.18a3.2	1	$( x^{3} + x + 1 )^{6} + 2 x^{2} ( x^{3} + x + 1 )^{2} + \left(2 x + 2\right) ( x^{3} + x + 1 ) + 2$	$A_4\wr C_2$ (as 18T113)	$288$	$2$	$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}]_{3}^{6}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{3}]_{3}^{6}$	$[\frac{4}{3},\frac{4}{3},\frac{4}{3}]^{2}$	$[\frac{1}{3},\frac{1}{3},\frac{1}{3}]^{2}$	$[1, 0]$	$[2, 1]$	$z^4 + z^2 + 1,z + t^2$	$[3, 7]$

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