Properties

Label 65.a
Number of curves $2$
Conductor $65$
CM no
Rank $1$
Graph

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

Elliptic curves in class 65.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
65.a1 65a1 \([1, 0, 0, -1, 0]\) \(117649/65\) \(65\) \([2]\) \(2\) \(-0.96162\) \(\Gamma_0(N)\)-optimal
65.a2 65a2 \([1, 0, 0, 4, 1]\) \(6967871/4225\) \(-4225\) \([2]\) \(4\) \(-0.61504\)  

Rank

sage: E.rank()
 

The elliptic curves in class 65.a have rank \(1\).

Complex multiplication

The elliptic curves in class 65.a do not have complex multiplication.

Modular form 65.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - 2 q^{3} - q^{4} - q^{5} + 2 q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{10} + 2 q^{11} + 2 q^{12} - q^{13} + 4 q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} - 6 q^{19} + O(q^{20})\) 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.