Properties

Label 690e
Number of curves $4$
Conductor $690$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 690e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
690.e3 690e1 \([1, 0, 1, -604, -5734]\) \(24310870577209/114462720\) \(114462720\) \([2]\) \(480\) \(0.39599\) \(\Gamma_0(N)\)-optimal
690.e2 690e2 \([1, 0, 1, -924, 922]\) \(87109155423289/49979073600\) \(49979073600\) \([2, 2]\) \(960\) \(0.74257\)  
690.e1 690e3 \([1, 0, 1, -10644, 420826]\) \(133345896593725369/340006815000\) \(340006815000\) \([2]\) \(1920\) \(1.0891\)  
690.e4 690e4 \([1, 0, 1, 3676, 8282]\) \(5495662324535111/3207841648920\) \(-3207841648920\) \([2]\) \(1920\) \(1.0891\)  

Rank

sage: E.rank()
 

The elliptic curves in class 690e have rank \(1\).

Complex multiplication

The elliptic curves in class 690e do not have complex multiplication.

Modular form 690.2.a.e

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.