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Properties

Label 8619.j
Number of curves $2$
Conductor $8619$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 8619.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8619.j1 8619j2 \([1, 0, 1, -16566, 629527]\) \(104154702625/24649677\) \(118979282790693\) \([2]\) \(21504\) \(1.4124\)  
8619.j2 8619j1 \([1, 0, 1, -5581, -152605]\) \(3981876625/232713\) \(1123261202817\) \([2]\) \(10752\) \(1.0658\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8619.j have rank \(1\).

Complex multiplication

The elliptic curves in class 8619.j do not have complex multiplication.

Modular form 8619.2.a.j

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