Properties

Label 64715e
Number of curves $3$
Conductor $64715$
CM no
Rank $1$
Graph

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

Elliptic curves in class 64715e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
64715.c2 64715e1 \([0, -1, 1, -2465, -51447]\) \(-262144/35\) \(-221247706715\) \([]\) \(51408\) \(0.90945\) \(\Gamma_0(N)\)-optimal
64715.c3 64715e2 \([0, -1, 1, 16025, 127906]\) \(71991296/42875\) \(-271028440725875\) \([]\) \(154224\) \(1.4588\)  
64715.c1 64715e3 \([0, -1, 1, -242835, 48262923]\) \(-250523582464/13671875\) \(-86424885435546875\) \([]\) \(462672\) \(2.0081\)  

Rank

sage: E.rank()
 

The elliptic curves in class 64715e have rank \(1\).

Complex multiplication

The elliptic curves in class 64715e do not have complex multiplication.

Modular form 64715.2.a.e

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

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.