LMFDB - $p$-adic family 2.7.2.21a

Defining polynomial over unramified subextension

$x^{2} + 4 b_{3} x + 8 c_{4} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$14$
Base field:	$\Q_{2}$
Ramification index $e$:	$2$
Residue field degree $f$:	$7$
Discriminant exponent $c$:	$21$
Artin slopes:	$[3]$
Swan slopes:	$[2]$
Means:	$\langle1\rangle$
Rams:	$(2)$
Field count:	$40$ (complete)
Ambiguity:	$14$
Mass:	$128$
Absolute Mass:	$128/7$

Diagrams

Varying

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

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 14
		$C_{14}$ (as 14T1)
hidden slopes	$[\ ]$	4

		Galois groups of order 112
		$C_2\times F_8$ (as 14T9)
hidden slopes	$[2,2,2]$	8

		Galois groups of order 896
		$C_2\wr C_7$ (as 14T29)
hidden slopes	$[2,2,2,2,2,2]$	28

Fields

Galois group	Galois Artin	Indices

Assoc. Inertia	Jump Set	Fields per packet

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Showing all 40

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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
2.7.2.21a1.1	4	$( x^{7} + x + 1 )^{2} + 2$	$C_{14}$ (as 14T1)	$14$	$14$	$[3]^{7}$	$[2]^{7}$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.2	4	$( x^{7} + x + 1 )^{2} + 10$	$C_{14}$ (as 14T1)	$14$	$14$	$[3]^{7}$	$[2]^{7}$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.3	28	$( x^{7} + x + 1 )^{2} + 4 x^{3} ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.4	28	$( x^{7} + x + 1 )^{2} + 4 x^{3} ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.5	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3}\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.6	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3}\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.7	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2}\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.8	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2}\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.9	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2}\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.10	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2}\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.11	8	$( x^{7} + x + 1 )^{2} + 4 x ( x^{7} + x + 1 ) + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.12	8	$( x^{7} + x + 1 )^{2} + 4 x ( x^{7} + x + 1 ) + 10$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.13	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.14	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.15	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.16	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.17	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4 x\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.18	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4 x\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.19	8	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x\right) ( x^{7} + x + 1 ) + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.20	8	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x\right) ( x^{7} + x + 1 ) + 10$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.21	4	$( x^{7} + x + 1 )^{2} + 4 ( x^{7} + x + 1 ) + 2$	$C_{14}$ (as 14T1)	$14$	$14$	$[3]^{7}$	$[2]^{7}$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.22	4	$( x^{7} + x + 1 )^{2} + 4 ( x^{7} + x + 1 ) + 10$	$C_{14}$ (as 14T1)	$14$	$14$	$[3]^{7}$	$[2]^{7}$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.23	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.24	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.25	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.26	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.27	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.28	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.29	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.30	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.31	8	$( x^{7} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.32	8	$( x^{7} + x + 1 )^{2} + \left(4 x + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.33	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.34	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.35	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.36	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.37	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4 x + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.38	28	$( x^{7} + x + 1 )^{2} + \left(4 x^{3} + 4 x^{2} + 4 x + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2 \wr C_7$ (as 14T29)	$896$	$2$	$[2, 2, 2, 2, 2, 2, 3]^{7}$	$[1,1,1,1,1,1,2]^{7}$	$[2,2,2,2,2,2]$	$[1,1,1,1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.39	8	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 4\right) ( x^{7} + x + 1 ) + 2$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$
2.7.2.21a1.40	8	$( x^{7} + x + 1 )^{2} + \left(4 x^{4} + 4 x^{3} + 4 x^{2} + 4 x + 4\right) ( x^{7} + x + 1 ) + 10$	$C_2\times F_8$ (as 14T9)	$112$	$2$	$[2, 2, 2, 3]^{7}$	$[1,1,1,2]^{7}$	$[2,2,2]$	$[1,1,1]$	$[2, 0]$	$[1]$	$z + 1$	$[1, 3]$

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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	2.7.2.21a
Base	2.1.1.0a1.1
Degree	\(14\)
e	\(2\)
f	\(7\)
c	\(21\)

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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.