Properties

Label 489762cw
Number of curves $2$
Conductor $489762$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("cw1")
 
E.isogeny_class()
 

Elliptic curves in class 489762cw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
489762.cw2 489762cw1 \([1, -1, 1, 4531, -520243]\) \(2924207/34776\) \(-122367833032536\) \([]\) \(1969920\) \(1.3841\) \(\Gamma_0(N)\)-optimal*
489762.cw1 489762cw2 \([1, -1, 1, -41099, 14647169]\) \(-2181825073/25039686\) \(-88108238889899046\) \([]\) \(5909760\) \(1.9334\) \(\Gamma_0(N)\)-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 489762cw1.

Rank

sage: E.rank()
 

The elliptic curves in class 489762cw have rank \(0\).

Complex multiplication

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

Modular form 489762.2.a.cw

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 3 q^{5} - q^{7} + q^{8} - 3 q^{10} - q^{14} + q^{16} - 8 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.