Properties

Label 402930dj
Number of curves $6$
Conductor $402930$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("402930.dj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 402930dj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
402930.dj5 402930dj1 [1, -1, 1, -920228, 1169881287] [2] 17694720 \(\Gamma_0(N)\)-optimal*
402930.dj4 402930dj2 [1, -1, 1, -23222948, 42991941831] [2, 2] 35389440 \(\Gamma_0(N)\)-optimal*
402930.dj1 402930dj3 [1, -1, 1, -371354468, 2754518724807] [2] 70778880 \(\Gamma_0(N)\)-optimal*
402930.dj3 402930dj4 [1, -1, 1, -31934948, 7809401031] [2, 2] 70778880  
402930.dj6 402930dj5 [1, -1, 1, 126841252, 62174371911] [2] 141557760  
402930.dj2 402930dj6 [1, -1, 1, -330103148, -2298342725049] [2] 141557760  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 402930dj1.

Rank

sage: E.rank()
 

The elliptic curves in class 402930dj have rank \(1\).

Modular form 402930.2.a.dj

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + 2q^{13} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.