$p$-adic family 3.1.12.19a

• Label: 3.1.12.19a
• Base: 3.1.1.0a1.1
• Degree: \(12\)
• e: \(12\)
• f: \(1\)
• c: \(19\)

Defining polynomial

$x^{12} + 3 b_{11} x^{11} + 3 b_{10} x^{10} + 3 a_{8} x^{8} + 3 d_{0} + 9 c_{12}$

Invariants

Residue field characteristic:	$3$
Degree:	$12$
Base field:	$\Q_{3}$
Ramification index $e$:	$12$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$19$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{2}{3}\rangle$
Rams:	$(4)$
Field count:	$48$ (complete)
Ambiguity:	$6$
Mass:	$18$
Absolute Mass:	$18$

Diagrams

Varying

Indices of inseparability:	$[8,0]$
Associated inertia:	$[2,1]$ (show 36), $[2,2]$ (show 12)
Jump Set:	undefined (show 24), $[2,6]$ (show 1), $[2,14]$ (show 6), $[2,16]$ (show 12), $[2,17]$ (show 3), $[2,18]$ (show 2)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 24
		$D_{12}$ (as 12T12)	$C_3\times D_4$ (as 12T14)	Total
hidden slopes	$[\ ]^{2}$	2	6	8

		Galois groups of order 72
		$C_3:D_{12}$ (as 12T38)	$C_6\wr C_2$ (as 12T42)	Total
hidden slopes	$[\frac{3}{2}]^{2}$	4	12	16

		Galois groups of order 216
		$C_3^2:D_{12}$ (as 12T118)	$S_3^2:C_6$ (as 12T121)	Total
hidden slopes	$[\frac{5}{4},\frac{5}{4}]^{2}$	2	6	8

		Galois groups of order 648
		$C_3\wr D_4$ (as 12T167)	$C_3^3:D_{12}$ (as 12T169)	Total
hidden slopes	$[\frac{5}{4},\frac{5}{4},\frac{3}{2}]^{2}$	12	4	16

Fields

Showing all 3

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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
3.1.12.19a2.4	6	$x^{12} + 3 x^{11} + 6 x^{8} + 3$	$S_3^2:C_6$ (as 12T121)	$216$	$3$	$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$	$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]^{2}$	$[\frac{1}{4},\frac{1}{4}]^{2}$	$[8, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 17]$
3.1.12.19a2.5	6	$x^{12} + 3 x^{11} + 6 x^{8} + 12$	$S_3^2:C_6$ (as 12T121)	$216$	$3$	$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$	$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]^{2}$	$[\frac{1}{4},\frac{1}{4}]^{2}$	$[8, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 17]$
3.1.12.19a2.6	6	$x^{12} + 3 x^{11} + 6 x^{8} + 21$	$S_3^2:C_6$ (as 12T121)	$216$	$3$	$[\frac{5}{4}, \frac{5}{4}, 2]_{4}^{2}$	$[\frac{1}{4},\frac{1}{4},1]_{4}^{2}$	$[\frac{5}{4},\frac{5}{4}]^{2}$	$[\frac{1}{4},\frac{1}{4}]^{2}$	$[8, 0]$	$[2, 1]$	$z^9 + z^6 + 1,z^2 + 2$	$[2, 17]$

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