Properties

Label 390390bf
Number of curves 4
Conductor 390390
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("390390.bf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 390390bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
390390.bf3 390390bf1 [1, 1, 0, -84672, -8269776] [2] 3538944 \(\Gamma_0(N)\)-optimal
390390.bf2 390390bf2 [1, 1, 0, -358452, 74247516] [2, 2] 7077888  
390390.bf1 390390bf3 [1, 1, 0, -5575482, 5064858414] [2] 14155776  
390390.bf4 390390bf4 [1, 1, 0, 478098, 370218906] [2] 14155776  

Rank

sage: E.rank()
 

The elliptic curves in class 390390bf have rank \(2\).

Modular form 390390.2.a.bf

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.