Properties

Label 489762dg
Number of curves $6$
Conductor $489762$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("489762.dg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 489762dg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
489762.dg5 489762dg1 [1, -1, 1, 191614, -66169983] [2] 9437184 \(\Gamma_0(N)\)-optimal*
489762.dg4 489762dg2 [1, -1, 1, -1755266, -765489279] [2, 2] 18874368 \(\Gamma_0(N)\)-optimal*
489762.dg3 489762dg3 [1, -1, 1, -7717586, 7510210881] [2, 2] 37748736 \(\Gamma_0(N)\)-optimal*
489762.dg2 489762dg4 [1, -1, 1, -26943026, -53820986943] [2] 37748736  
489762.dg1 489762dg5 [1, -1, 1, -120362846, 508285978737] [2] 75497472 \(\Gamma_0(N)\)-optimal*
489762.dg6 489762dg6 [1, -1, 1, 9530554, 36307705425] [2] 75497472  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 489762dg1.

Rank

sage: E.rank()
 

The elliptic curves in class 489762dg have rank \(2\).

Modular form 489762.2.a.dg

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.