Properties

Label 468270.bf
Number of curves $4$
Conductor $468270$
CM no
Rank $1$
Graph

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

Elliptic curves in class 468270.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
468270.bf1 468270bf3 [1, -1, 0, -913149, 333800055] [2] 8847360 \(\Gamma_0(N)\)-optimal*
468270.bf2 468270bf2 [1, -1, 0, -96399, -2864295] [2, 2] 4423680 \(\Gamma_0(N)\)-optimal*
468270.bf3 468270bf1 [1, -1, 0, -74619, -7817067] [2] 2211840 \(\Gamma_0(N)\)-optimal*
468270.bf4 468270bf4 [1, -1, 0, 371871, -22812597] [2] 8847360  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 468270.bf1.

Rank

sage: E.rank()
 

The elliptic curves in class 468270.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 468270.bf do not have complex multiplication.

Modular form 468270.2.a.bf

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} + q^{5} - 4q^{7} - q^{8} - q^{10} + 2q^{13} + 4q^{14} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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 LMFDB 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 LMFDB labels.