LMFDB - $p$-adic family 2.6.2.12a

Defining polynomial over unramified subextension

$x^{2} + 2 a_{1} x + 4 c_{2} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$12$
Base field:	$\Q_{2}$
Ramification index $e$:	$2$
Residue field degree $f$:	$6$
Discriminant exponent $c$:	$12$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(1)$
Field count:	$26$ (complete)
Ambiguity:	$12$
Mass:	$63$
Absolute Mass:	$21/2$

Diagrams

Varying

Indices of inseparability:	$[1,0]$
Associated inertia:	$[1]$
Jump Set:	$[1,2]$ (show 1), $[1,3]$ (show 24), $[1,4]$ (show 1)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 12
		$C_{12}$ (as 12T1)	$C_2\times C_6$ (as 12T2)	Total
hidden slopes	$[\ ]$	1	1	2

		Galois groups of order 24
		$C_2\times A_4$ (as 12T7)	$C_3\times D_4$ (as 12T14)	Total
hidden slopes	$[2]$	1	2	3

		Galois groups of order 48
		$C_2^2\times A_4$ (as 12T25)	$C_4\times A_4$ (as 12T29)	Total
hidden slopes	$[2,2]$	1	1	2

$[2]^{2}$		1	1

Total	1	2	3

		Galois groups of order 96
		$D_4\times A_4$ (as 12T51)	$C_2^3:A_4$ (as 12T58)	Total
hidden slopes	$[2,2,2]$	2	2	4

		Galois groups of order 192
		$C_2^4:A_4$ (as 12T87)	$C_2^4:C_{12}$ (as 12T105)	Total
hidden slopes	$[2,2,2,2]$	2	2	4

$[2,2,2]^{2}$		2	2

Total	2	4	6

		Galois groups of order 384
		$C_2\wr C_6$ (as 12T134)
hidden slopes	$[2,2,2,2,2]$	8

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set	Fields per packet

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Showing all 26

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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
2.6.2.12a1.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_6\times C_2$ (as 12T2)	$12$	$12$	$[2]^{6}$	$[1]^{6}$	$[\ ]$	$[\ ]$	$[1, 0]$	$[1]$	$z + (t^2 + 1)$	$[1, 2]$
2.6.2.12a1.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x^{3} + 2$	$C_{12}$ (as 12T1)	$12$	$12$	$[2]^{6}$	$[1]^{6}$	$[\ ]$	$[\ ]$	$[1, 0]$	$[1]$	$z + (t^2 + 1)$	$[1, 4]$
2.6.2.12a2.1	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 x ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$D_4\times A_4$ (as 12T51)	$96$	$2$	$[2, 2, 2, 2]^{6}$	$[1,1,1,1]^{6}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t)$	$[1, 3]$
2.6.2.12a2.2	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 x ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$	$D_4\times A_4$ (as 12T51)	$96$	$2$	$[2, 2, 2, 2]^{6}$	$[1,1,1,1]^{6}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t)$	$[1, 3]$
2.6.2.12a3.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 x^{3} ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2^3:A_4$ (as 12T58)	$96$	$2$	$[2, 2, 2, 2]^{6}$	$[1,1,1,1]^{6}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^3)$	$[1, 3]$
2.6.2.12a3.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 x^{3} ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x + 2$	$C_2^4:C_{12}$ (as 12T105)	$192$	$2$	$[2, 2, 2, 2]^{12}$	$[1,1,1,1]^{12}$	$[2,2,2]^{2}$	$[1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^5 + t^3)$	$[1, 3]$
2.6.2.12a4.1	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^3 + t^2 + 1)$	$[1, 3]$
2.6.2.12a4.2	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^3 + t^2 + 1)$	$[1, 3]$
2.6.2.12a5.1	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2} + 2 x\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$D_4 \times C_3$ (as 12T14)	$24$	$6$	$[2, 2]^{6}$	$[1,1]^{6}$	$[2]$	$[1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4 + t^2 + t)$	$[1, 3]$
2.6.2.12a5.2	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2} + 2 x\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$	$D_4 \times C_3$ (as 12T14)	$24$	$6$	$[2, 2]^{6}$	$[1,1]^{6}$	$[2]$	$[1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4 + t^2 + t)$	$[1, 3]$
2.6.2.12a6.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2^4:A_4$ (as 12T87)	$192$	$2$	$[2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1]^{6}$	$[2,2,2,2]$	$[1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^2 + t)$	$[1, 3]$
2.6.2.12a6.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x^{2} + 2$	$C_2^4:C_{12}$ (as 12T105)	$192$	$2$	$[2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1]^{6}$	$[2,2,2,2]$	$[1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^2 + t)$	$[1, 3]$
2.6.2.12a7.1	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 x^{5} ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^4 + t^2 + t)$	$[1, 3]$
2.6.2.12a7.2	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + 2 x^{5} ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^4 + t^2 + t)$	$[1, 3]$
2.6.2.12a8.1	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{2}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.6.2.12a8.2	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{2}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.6.2.12a9.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{2}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$A_4 \times C_2$ (as 12T7)	$24$	$4$	$[2, 2]^{6}$	$[1,1]^{6}$	$[2]$	$[1]$	$[1, 0]$	$[1]$	$z + (t^3 + 1)$	$[1, 3]$
2.6.2.12a9.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{2}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x^{2} + 2$	$C_4\times A_4$ (as 12T29)	$48$	$4$	$[2, 2]^{12}$	$[1,1]^{12}$	$[2]^{2}$	$[1]^{2}$	$[1, 0]$	$[1]$	$z + (t^3 + 1)$	$[1, 3]$
2.6.2.12a10.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{2} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2^2 \times A_4$ (as 12T25)	$48$	$4$	$[2, 2, 2]^{6}$	$[1,1,1]^{6}$	$[2,2]$	$[1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t^2)$	$[1, 3]$
2.6.2.12a10.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{2} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x + 2$	$C_4\times A_4$ (as 12T29)	$48$	$4$	$[2, 2, 2]^{6}$	$[1,1,1]^{6}$	$[2,2]$	$[1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t^2)$	$[1, 3]$
2.6.2.12a11.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{2} + 2 x\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2^3:A_4$ (as 12T58)	$96$	$2$	$[2, 2, 2, 2]^{6}$	$[1,1,1,1]^{6}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + (t + 1)$	$[1, 3]$
2.6.2.12a11.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{2} + 2 x\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x + 2$	$C_2^4:C_{12}$ (as 12T105)	$192$	$2$	$[2, 2, 2, 2]^{12}$	$[1,1,1,1]^{12}$	$[2,2,2]^{2}$	$[1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t + 1)$	$[1, 3]$
2.6.2.12a12.1	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2^4:C_{12}$ (as 12T105)	$192$	$2$	$[2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1]^{6}$	$[2,2,2,2]$	$[1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4 + t^2 + 1)$	$[1, 3]$
2.6.2.12a12.2	1	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3}\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 4 x^{2} + 2$	$C_2^4:A_4$ (as 12T87)	$192$	$2$	$[2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1]^{6}$	$[2,2,2,2]$	$[1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4 + t^2 + 1)$	$[1, 3]$
2.6.2.12a13.1	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 2$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4)$	$[1, 3]$
2.6.2.12a13.2	2	$( x^{6} + x^{4} + x^{3} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3} + 2\right) ( x^{6} + x^{4} + x^{3} + x + 1 ) + 6$	$C_2\wr C_6$ (as 12T134)	$384$	$2$	$[2, 2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1,1]^{6}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4)$	$[1, 3]$

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Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Label	2.6.2.12a
Base	2.1.1.0a1.1
Degree	\(12\)
e	\(2\)
f	\(6\)
c	\(12\)

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Defining polynomial over unramified subextension

Invariants

Diagrams

Varying

Galois groups and Hidden Artin slopes

Fields

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.