Properties

Label 64009b
Number of curves $2$
Conductor $64009$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 64009b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
64009.c2 64009b1 [1, 1, 0, -1333, 72714] [] 75504 \(\Gamma_0(N)\)-optimal
64009.c1 64009b2 [1, 1, 0, -1921603, -1026119574] [] 830544  

Rank

sage: E.rank()
 

The elliptic curves in class 64009b have rank \(1\).

Complex multiplication

The elliptic curves in class 64009b do not have complex multiplication.

Modular form 64009.2.a.b

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

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.

\(\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.