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Properties

Label 482664bi
Number of curves $2$
Conductor $482664$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bi1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 482664bi have rank \(0\).

Complex multiplication

The elliptic curves in class 482664bi do not have complex multiplication.

Modular form 482664.2.a.bi

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 482664bi

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
482664.bi2 482664bi1 \([0, 1, 0, -1019464, 4134262016]\) \(-23707171994692/1480419781911\) \(-7317200411756597246976\) \([2]\) \(36495360\) \(2.8752\) \(\Gamma_0(N)\)-optimal*
482664.bi1 482664bi2 \([0, 1, 0, -45371824, 116860220192]\) \(1044942448578893426/7759962920241\) \(76709598951599187904512\) \([2]\) \(72990720\) \(3.2218\) \(\Gamma_0(N)\)-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 482664bi1.