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Properties

Label 78650bz
Number of curves $2$
Conductor $78650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bz1")
 
E.isogeny_class()
 

Elliptic curves in class 78650bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
78650.dj2 78650bz1 \([1, -1, 1, -8130, -429503]\) \(-2146689/1664\) \(-46060586000000\) \([]\) \(392000\) \(1.3205\) \(\Gamma_0(N)\)-optimal
78650.dj1 78650bz2 \([1, -1, 1, -643380, 218096497]\) \(-1064019559329/125497034\) \(-3473838297657406250\) \([]\) \(2744000\) \(2.2935\)  

Rank

sage: E.rank()
 

The elliptic curves in class 78650bz have rank \(1\).

Complex multiplication

The elliptic curves in class 78650bz do not have complex multiplication.

Modular form 78650.2.a.bz

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.