Properties

Label 60690bj
Number of curves 8
Conductor 60690
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("60690.bk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 60690bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
60690.bk7 60690bj1 [1, 1, 1, 60684, 4361013] [2] 655360 \(\Gamma_0(N)\)-optimal
60690.bk6 60690bj2 [1, 1, 1, -309236, 38689589] [2, 2] 1310720  
60690.bk5 60690bj3 [1, 1, 1, -2181956, -1213785547] [2, 2] 2621440  
60690.bk4 60690bj4 [1, 1, 1, -4355236, 3495591989] [2] 2621440  
60690.bk8 60690bj5 [1, 1, 1, 367024, -3876959851] [2] 5242880  
60690.bk2 60690bj6 [1, 1, 1, -34694456, -78671565547] [2, 2] 5242880  
60690.bk3 60690bj7 [1, 1, 1, -34477706, -79702775347] [2] 10485760  
60690.bk1 60690bj8 [1, 1, 1, -555111206, -5034288025747] [2] 10485760  

Rank

sage: E.rank()
 

The elliptic curves in class 60690bj have rank \(1\).

Modular form 60690.2.a.bk

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.