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Properties

Label 265650.a
Number of curves $2$
Conductor $265650$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 265650.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
265650.a1 265650a2 \([1, 1, 0, -1551700, 743331250]\) \(26444015547214434625/46191222\) \(721737843750\) \([2]\) \(3096576\) \(1.9633\)  
265650.a2 265650a1 \([1, 1, 0, -96950, 11592000]\) \(-6449916994998625/8532911772\) \(-133326746437500\) \([2]\) \(1548288\) \(1.6167\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 265650.a have rank \(2\).

Complex multiplication

The elliptic curves in class 265650.a do not have complex multiplication.

Modular form 265650.2.a.a

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