Properties

Label 665d
Number of curves $2$
Conductor $665$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 665d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
665.a2 665d1 \([0, -1, 1, -210, 6798]\) \(-1029077364736/18960396875\) \(-18960396875\) \([5]\) \(600\) \(0.65320\) \(\Gamma_0(N)\)-optimal
665.a1 665d2 \([0, -1, 1, -16660, -1081562]\) \(-511416541770305536/214587319023035\) \(-214587319023035\) \([]\) \(3000\) \(1.4579\)  

Rank

sage: E.rank()
 

The elliptic curves in class 665d have rank \(1\).

Complex multiplication

The elliptic curves in class 665d do not have complex multiplication.

Modular form 665.2.a.d

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.