Properties

Label 4290bb
Number of curves $8$
Conductor $4290$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 4290bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4290.bb7 4290bb1 \([1, 0, 0, -11420, -450288]\) \(164711681450297281/8097103872000\) \(8097103872000\) \([6]\) \(13824\) \(1.2362\) \(\Gamma_0(N)\)-optimal
4290.bb6 4290bb2 \([1, 0, 0, -31900, 1610000]\) \(3590017885052913601/954068544000000\) \(954068544000000\) \([2, 6]\) \(27648\) \(1.5828\)  
4290.bb3 4290bb3 \([1, 0, 0, -913820, -336308208]\) \(84392862605474684114881/11228954880\) \(11228954880\) \([2]\) \(41472\) \(1.7855\)  
4290.bb5 4290bb4 \([1, 0, 0, -471900, 124722000]\) \(11621808143080380273601/1335706803288000\) \(1335706803288000\) \([12]\) \(55296\) \(1.9294\)  
4290.bb8 4290bb5 \([1, 0, 0, 80420, 10438352]\) \(57519563401957999679/80296734375000000\) \(-80296734375000000\) \([6]\) \(55296\) \(1.9294\)  
4290.bb2 4290bb6 \([1, 0, 0, -913900, -336246400]\) \(84415028961834287121601/30783551683856400\) \(30783551683856400\) \([2, 2]\) \(82944\) \(2.1321\)  
4290.bb1 4290bb7 \([1, 0, 0, -1047000, -231922620]\) \(126929854754212758768001/50235797102795981820\) \(50235797102795981820\) \([4]\) \(165888\) \(2.4787\)  
4290.bb4 4290bb8 \([1, 0, 0, -782080, -436614148]\) \(-52902632853833942200321/51713453577420277500\) \(-51713453577420277500\) \([2]\) \(165888\) \(2.4787\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4290bb have rank \(1\).

Complex multiplication

The elliptic curves in class 4290bb do not have complex multiplication.

Modular form 4290.2.a.bb

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 4 q^{7} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + q^{13} - 4 q^{14} + q^{15} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.