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Properties

Label 858.j
Number of curves $2$
Conductor $858$
CM no
Rank $0$
Graph

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

Elliptic curves in class 858.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
858.j1 858j2 \([1, 0, 0, -617, -5961]\) \(-25979045828113/52635726\) \(-52635726\) \([]\) \(720\) \(0.36849\)  
858.j2 858j1 \([1, 0, 0, 13, -39]\) \(241804367/833976\) \(-833976\) \([3]\) \(240\) \(-0.18082\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 858.j have rank \(0\).

Complex multiplication

The elliptic curves in class 858.j do not have complex multiplication.

Modular form 858.2.a.j

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 3 q^{5} + q^{6} + 5 q^{7} + q^{8} + q^{9} - 3 q^{10} - q^{11} + q^{12} + q^{13} + 5 q^{14} - 3 q^{15} + q^{16} + q^{18} + 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.