Properties

Label 491970.bd
Number of curves $2$
Conductor $491970$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bd1")
 
E.isogeny_class()
 

Elliptic curves in class 491970.bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
491970.bd1 491970bd2 \([1, 0, 1, -1064624, 148057166]\) \(901456690969801/457629750000\) \(67745626874097750000\) \([2]\) \(21626880\) \(2.4977\) \(\Gamma_0(N)\)-optimal*
491970.bd2 491970bd1 \([1, 0, 1, 247296, 17914702]\) \(11298232190519/7472736000\) \(-1106233117022304000\) \([2]\) \(10813440\) \(2.1511\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 491970.bd1.

Rank

sage: E.rank()
 

The elliptic curves in class 491970.bd have rank \(1\).

Complex multiplication

The elliptic curves in class 491970.bd do not have complex multiplication.

Modular form 491970.2.a.bd

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} - 4 q^{13} - 2 q^{14} - q^{15} + q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.