Properties

Label 489762.bl
Number of curves $2$
Conductor $489762$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands for: SageMath
sage: E = EllipticCurve("bl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 489762.bl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
489762.bl1 489762bl2 \([1, -1, 0, -920997, -318266987]\) \(24553362849625/1755162752\) \(6175967983139590272\) \([2]\) \(12644352\) \(2.3525\) \(\Gamma_0(N)\)-optimal*
489762.bl2 489762bl1 \([1, -1, 0, 52443, -21757163]\) \(4533086375/60669952\) \(-213482015080169472\) \([2]\) \(6322176\) \(2.0060\) \(\Gamma_0(N)\)-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 489762.bl1.

Rank

sage: E.rank()
 

The elliptic curves in class 489762.bl have rank \(1\).

Complex multiplication

The elliptic curves in class 489762.bl do not have complex multiplication.

Modular form 489762.2.a.bl

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