Properties

Label 481650.gq
Number of curves $4$
Conductor $481650$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("481650.gq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 481650.gq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
481650.gq1 481650gq3 [1, 1, 1, -12844088, 17712172031] [2] 21233664 \(\Gamma_0(N)\)-optimal*
481650.gq2 481650gq4 [1, 1, 1, -929588, 183154031] [2] 21233664  
481650.gq3 481650gq2 [1, 1, 1, -802838, 276442031] [2, 2] 10616832 \(\Gamma_0(N)\)-optimal*
481650.gq4 481650gq1 [1, 1, 1, -42338, 5704031] [2] 5308416 \(\Gamma_0(N)\)-optimal*
*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 481650.gq4.

Rank

sage: E.rank()
 

The elliptic curves in class 481650.gq have rank \(1\).

Modular form 481650.2.a.gq

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.