Properties

Label 8640.g
Number of curves $2$
Conductor $8640$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("g1")
 
E.isogeny_class()
 

Elliptic curves in class 8640.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8640.g1 8640bw2 \([0, 0, 0, -1188, 15768]\) \(-9199872/5\) \(-100776960\) \([]\) \(3456\) \(0.48440\)  
8640.g2 8640bw1 \([0, 0, 0, 12, 88]\) \(6912/125\) \(-3456000\) \([]\) \(1152\) \(-0.064910\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8640.g have rank \(1\).

Complex multiplication

The elliptic curves in class 8640.g do not have complex multiplication.

Modular form 8640.2.a.g

sage: E.q_eigenform(10)
 
\(q - q^{5} - 2 q^{7} - 2 q^{13} - 3 q^{17} + 5 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.