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Properties

Label 8619.k
Number of curves $2$
Conductor $8619$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 8619.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8619.k1 8619m2 \([1, 0, 1, -56, -115]\) \(8615125/2601\) \(5714397\) \([2]\) \(1536\) \(0.0016982\)  
8619.k2 8619m1 \([1, 0, 1, 9, -11]\) \(42875/51\) \(-112047\) \([2]\) \(768\) \(-0.34488\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8619.k have rank \(0\).

Complex multiplication

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

Modular form 8619.2.a.k

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