Properties

Label 210.c
Number of curves 66
Conductor 210210
CM no
Rank 00
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("c1") E.isogeny_class()
 

Elliptic curves in class 210.c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210.c1 210c5 [1,1,1,16800,845133][1, 1, 1, -16800, -845133] 524388516989299201/3150524388516989299201/3150 31503150 [2][2] 256256 0.736760.73676  
210.c2 210c3 [1,1,1,1050,13533][1, 1, 1, -1050, -13533] 128031684631201/9922500128031684631201/9922500 99225009922500 [2,2][2, 2] 128128 0.390180.39018  
210.c3 210c6 [1,1,1,980,15325][1, 1, 1, -980, -15325] 104094944089921/35880468750-104094944089921/35880468750 35880468750-35880468750 [2][2] 256256 0.736760.73676  
210.c4 210c4 [1,1,1,370,2435][1, 1, 1, -370, 2435] 5602762882081/3458880605602762882081/345888060 345888060345888060 [4][4] 128128 0.390180.39018  
210.c5 210c2 [1,1,1,70,205][1, 1, 1, -70, -205] 37966934881/864360037966934881/8643600 86436008643600 [2,4][2, 4] 6464 0.0436100.043610  
210.c6 210c1 [1,1,1,10,13][1, 1, 1, 10, -13] 109902239/188160109902239/188160 188160-188160 [4][4] 3232 0.30296-0.30296 Γ0(N)\Gamma_0(N)-optimal

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 210.c have rank 00.

L-function data

 
Bad L-factors:
Prime L-Factor
221T1 - T
331+T1 + T
551T1 - T
771T1 - T
 
Good L-factors:
Prime L-Factor Isogeny Class over Fp\mathbb{F}_p
1111 14T+11T2 1 - 4 T + 11 T^{2} 1.11.ae
1313 1+2T+13T2 1 + 2 T + 13 T^{2} 1.13.c
1717 12T+17T2 1 - 2 T + 17 T^{2} 1.17.ac
1919 1+4T+19T2 1 + 4 T + 19 T^{2} 1.19.e
2323 1+8T+23T2 1 + 8 T + 23 T^{2} 1.23.i
2929 16T+29T2 1 - 6 T + 29 T^{2} 1.29.ag
\cdots\cdots\cdots
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 210.c do not have complex multiplication.

Modular form 210.2.a.c

Copy content sage:E.q_eigenform(10)
 
q+q2q3+q4+q5q6+q7+q8+q9+q10+4q11q122q13+q14q15+q16+2q17+q184q19+O(q20)q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{7} + q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - q^{15} + q^{16} + 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the LMFDB numbering.

(124848212424421848848124424212848421)\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.