Properties

Label 4641.e
Number of curves $2$
Conductor $4641$
CM no
Rank $1$
Graph

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

Elliptic curves in class 4641.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4641.e1 4641d1 \([1, 0, 1, -46, 107]\) \(10431681625/710073\) \(710073\) \([2]\) \(768\) \(-0.12872\) \(\Gamma_0(N)\)-optimal
4641.e2 4641d2 \([1, 0, 1, 39, 481]\) \(6804992375/102626433\) \(-102626433\) \([2]\) \(1536\) \(0.21785\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4641.e have rank \(1\).

Complex multiplication

The elliptic curves in class 4641.e do not have complex multiplication.

Modular form 4641.2.a.e

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