Properties

Label 494190bn
Number of curves $4$
Conductor $494190$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("494190.bn1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 494190bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
494190.bn4 494190bn1 [1, -1, 0, -26064, -2757200] [2] 3932160 \(\Gamma_0(N)\)-optimal*
494190.bn3 494190bn2 [1, -1, 0, -494244, -133566692] [2, 2] 7864320 \(\Gamma_0(N)\)-optimal*
494190.bn2 494190bn3 [1, -1, 0, -572274, -88512170] [2] 15728640 \(\Gamma_0(N)\)-optimal*
494190.bn1 494190bn4 [1, -1, 0, -7907094, -8556046862] [2] 15728640  
*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 494190bn1.

Rank

sage: E.rank()
 

The elliptic curves in class 494190bn have rank \(1\).

Modular form 494190.2.a.bn

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

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 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.