Properties

Label 840.a
Number of curves $4$
Conductor $840$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 840.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
840.a1 840a3 \([0, -1, 0, -1736, 26796]\) \(282678688658/18600435\) \(38093690880\) \([2]\) \(768\) \(0.77990\)  
840.a2 840a2 \([0, -1, 0, -336, -1764]\) \(4108974916/893025\) \(914457600\) \([2, 2]\) \(384\) \(0.43333\)  
840.a3 840a1 \([0, -1, 0, -316, -2060]\) \(13674725584/945\) \(241920\) \([2]\) \(192\) \(0.086753\) \(\Gamma_0(N)\)-optimal
840.a4 840a4 \([0, -1, 0, 744, -11700]\) \(22208984782/40516875\) \(-82978560000\) \([2]\) \(768\) \(0.77990\)  

Rank

sage: E.rank()
 

The elliptic curves in class 840.a have rank \(1\).

Complex multiplication

The elliptic curves in class 840.a do not have complex multiplication.

Modular form 840.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + 4 q^{11} - 2 q^{13} + q^{15} - 6 q^{17} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.