LMFDB - $p$-adic family 2.7.2.14a

Defining polynomial over unramified subextension

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

Invariants

Residue field characteristic:	$2$
Degree:	$14$
Base field:	$\Q_{2}$
Ramification index $e$:	$2$
Residue field degree $f$:	$7$
Discriminant exponent $c$:	$14$
Artin slopes:	$[2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(1)$
Field count:	$38$ (complete)
Ambiguity:	$14$
Mass:	$127$
Absolute Mass:	$127/7$

Diagrams

Varying

Indices of inseparability:	$[1,0]$
Associated inertia:	$[1]$
Jump Set:	$[1,2]$ (show 1), $[1,3]$ (show 36), $[1,4]$ (show 1)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 14
		$C_{14}$ (as 14T1)
hidden slopes	$[\ ]$	2

		Galois groups of order 56
		$F_8$ (as 14T6)
hidden slopes	$[2,2]$	2

		Galois groups of order 112
		$C_2\times F_8$ (as 14T9)
hidden slopes	$[2,2,2]$	4

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

Total	6

		Galois groups of order 448
		$C_2^3:F_8$ (as 14T21)
hidden slopes	$[2,2,2,2,2]$	7

		Galois groups of order 896
		$C_2\wr C_7$ (as 14T29)
hidden slopes	$[2,2,2,2,2,2]$	14

$[2,2,2,2,2]^{2}$	7

Total	21

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set	Fields per packet

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Showing all 38

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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
2.7.2.14a1.1	2	$( x^{7} + x + 1 )^{2} + 2 ( x^{7} + x + 1 ) + 2$	$C_{14}$ (as 14T1)	$14$	$14$	$[2]^{7}$	$[1]^{7}$	$[\ ]$	$[\ ]$	$[1, 0]$	$[1]$	$z + 1$	$[1, 2]$
2.7.2.14a1.2	2	$( x^{7} + x + 1 )^{2} + 2 ( x^{7} + x + 1 ) + 6$	$C_{14}$ (as 14T1)	$14$	$14$	$[2]^{7}$	$[1]^{7}$	$[\ ]$	$[\ ]$	$[1, 0]$	$[1]$	$z + 1$	$[1, 4]$
2.7.2.14a2.1	2	$( x^{7} + x + 1 )^{2} + 2 x ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.7.2.14a2.2	2	$( x^{7} + x + 1 )^{2} + 2 x ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + t$	$[1, 3]$
2.7.2.14a3.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t + 1)$	$[1, 3]$
2.7.2.14a3.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x + 2\right) ( x^{7} + x + 1 ) + 4 x^{2} + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t + 1)$	$[1, 3]$
2.7.2.14a4.1	2	$( x^{7} + x + 1 )^{2} + 2 x^{3} ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + t^3$	$[1, 3]$
2.7.2.14a4.2	2	$( x^{7} + x + 1 )^{2} + 2 x^{3} ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + t^3$	$[1, 3]$
2.7.2.14a5.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2\right) ( x^{7} + x + 1 ) + 2$	$F_8$ (as 14T6)	$56$	$2$	$[2, 2, 2]^{7}$	$[1,1,1]^{7}$	$[2,2]$	$[1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + 1)$	$[1, 3]$
2.7.2.14a5.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2\right) ( x^{7} + x + 1 ) + 4 x^{6} + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2]^{14}$	$[1,1,1]^{14}$	$[2,2]^{2}$	$[1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^3 + 1)$	$[1, 3]$
2.7.2.14a6.1	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x\right) ( x^{7} + x + 1 ) + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 2]^{7}$	$[1,1,1,1]^{7}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t)$	$[1, 3]$
2.7.2.14a6.2	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x\right) ( x^{7} + x + 1 ) + 6$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 2]^{7}$	$[1,1,1,1]^{7}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t)$	$[1, 3]$
2.7.2.14a7.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t + 1)$	$[1, 3]$
2.7.2.14a7.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x + 2\right) ( x^{7} + x + 1 ) + 4 x^{2} + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^3 + t + 1)$	$[1, 3]$
2.7.2.14a8.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2}\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^3 + t^2)$	$[1, 3]$
2.7.2.14a8.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2}\right) ( x^{7} + x + 1 ) + 4 x^{2} + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t^2)$	$[1, 3]$
2.7.2.14a9.1	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t^2 + t + 1)$	$[1, 3]$
2.7.2.14a9.2	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^3 + t^2 + t + 1)$	$[1, 3]$
2.7.2.14a10.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^4 + t^3 + t^2 + t + 1)$	$[1, 3]$
2.7.2.14a10.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 2\right) ( x^{7} + x + 1 ) + 4 x^{2} + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^4 + t^3 + t^2 + t + 1)$	$[1, 3]$
2.7.2.14a11.1	2	$( x^{7} + x + 1 )^{2} + 2 x^{5} ( x^{7} + x + 1 ) + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 2]^{7}$	$[1,1,1,1]^{7}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + t^5$	$[1, 3]$
2.7.2.14a11.2	2	$( x^{7} + x + 1 )^{2} + 2 x^{5} ( x^{7} + x + 1 ) + 6$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 2]^{7}$	$[1,1,1,1]^{7}$	$[2,2,2]$	$[1,1,1]$	$[1, 0]$	$[1]$	$z + t^5$	$[1, 3]$
2.7.2.14a12.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x\right) ( x^{7} + x + 1 ) + 2$	$F_8$ (as 14T6)	$56$	$2$	$[2, 2, 2]^{7}$	$[1,1,1]^{7}$	$[2,2]$	$[1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t)$	$[1, 3]$
2.7.2.14a12.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x\right) ( x^{7} + x + 1 ) + 4 x + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2]^{14}$	$[1,1,1]^{14}$	$[2,2]^{2}$	$[1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^5 + t)$	$[1, 3]$
2.7.2.14a13.1	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t + 1)$	$[1, 3]$
2.7.2.14a13.2	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x + 2\right) ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t + 1)$	$[1, 3]$
2.7.2.14a14.1	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{3} + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^3 + 1)$	$[1, 3]$
2.7.2.14a14.2	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{3} + 2\right) ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^3 + 1)$	$[1, 3]$
2.7.2.14a15.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4}\right) ( x^{7} + x + 1 ) + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4)$	$[1, 3]$
2.7.2.14a15.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4}\right) ( x^{7} + x + 1 ) + 4 x + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^5 + t^4)$	$[1, 3]$
2.7.2.14a16.1	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2}\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4 + t^3 + t^2)$	$[1, 3]$
2.7.2.14a16.2	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2}\right) ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^5 + t^4 + t^3 + t^2)$	$[1, 3]$
2.7.2.14a17.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{6} + 2 x\right) ( x^{7} + x + 1 ) + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^6 + t)$	$[1, 3]$
2.7.2.14a17.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{6} + 2 x\right) ( x^{7} + x + 1 ) + 4 x^{2} + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^6 + t)$	$[1, 3]$
2.7.2.14a18.1	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{6} + 2 x + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2]^{14}$	$[1,1,1,1,1,1]^{14}$	$[2,2,2,2,2]^{2}$	$[1,1,1,1,1]^{2}$	$[1, 0]$	$[1]$	$z + (t^6 + t + 1)$	$[1, 3]$
2.7.2.14a18.2	1	$( x^{7} + x + 1 )^{2} + \left(2 x^{6} + 2 x + 2\right) ( x^{7} + x + 1 ) + 4 x + 2$	$C_2^3:F_8$ (as 14T21)	$448$	$2$	$[2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1]^{7}$	$[2,2,2,2,2]$	$[1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^6 + t + 1)$	$[1, 3]$
2.7.2.14a19.1	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{6} + 2 x^{3} + 2\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^6 + t^3 + 1)$	$[1, 3]$
2.7.2.14a19.2	2	$( x^{7} + x + 1 )^{2} + \left(2 x^{6} + 2 x^{3} + 2\right) ( x^{7} + x + 1 ) + 6$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 2]^{7}$	$[1,1,1,1,1,1,1]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[1, 0]$	$[1]$	$z + (t^6 + t^3 + 1)$	$[1, 3]$

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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	2.7.2.14a
Base	2.1.1.0a1.1
Degree	\(14\)
e	\(2\)
f	\(7\)
c	\(14\)

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Defining polynomial over unramified subextension

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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.