Properties

Label 210210da
Number of curves $8$
Conductor $210210$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("da1")
 
E.isogeny_class()
 

Elliptic curves in class 210210da

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.cs7 210210da1 \([1, 1, 1, -559581, 153889203]\) \(164711681450297281/8097103872000\) \(952616173436928000\) \([2]\) \(3981312\) \(2.2092\) \(\Gamma_0(N)\)-optimal
210210.cs6 210210da2 \([1, 1, 1, -1563101, -553793101]\) \(3590017885052913601/954068544000000\) \(112245210133056000000\) \([2, 2]\) \(7962624\) \(2.5557\)  
210210.cs3 210210da3 \([1, 1, 1, -44777181, 115308938163]\) \(84392862605474684114881/11228954880\) \(1321075312677120\) \([2]\) \(11943936\) \(2.7585\)  
210210.cs8 210210da4 \([1, 1, 1, 3940579, -3576414157]\) \(57519563401957999679/80296734375000000\) \(-9446830502484375000000\) \([2]\) \(15925248\) \(2.9023\)  
210210.cs5 210210da5 \([1, 1, 1, -23123101, -42802769101]\) \(11621808143080380273601/1335706803288000\) \(157144569700029912000\) \([2]\) \(15925248\) \(2.9023\)  
210210.cs2 210210da6 \([1, 1, 1, -44781101, 115287734099]\) \(84415028961834287121601/30783551683856400\) \(3621654072054021603600\) \([2, 2]\) \(23887872\) \(3.1050\)  
210210.cs4 210210da7 \([1, 1, 1, -38321921, 149720330843]\) \(-52902632853833942200321/51713453577420277500\) \(-6084036099929918227597500\) \([2]\) \(47775744\) \(3.4516\)  
210210.cs1 210210da8 \([1, 1, 1, -51303001, 79498155659]\) \(126929854754212758768001/50235797102795981820\) \(5910191293346844465141180\) \([2]\) \(47775744\) \(3.4516\)  

Rank

sage: E.rank()
 

The elliptic curves in class 210210da have rank \(1\).

Complex multiplication

The elliptic curves in class 210210da do not have complex multiplication.

Modular form 210210.2.a.da

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} + q^{8} + q^{9} - q^{10} - q^{11} - q^{12} - q^{13} + q^{15} + q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.