Properties

Label 73920.er
Number of curves $6$
Conductor $73920$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("er1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 73920.er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
73920.er1 73920gd6 [0, 1, 0, -195148801, 1049228361215] [2] 3932160  
73920.er2 73920gd4 [0, 1, 0, -12196801, 16391140415] [2, 2] 1966080  
73920.er3 73920gd5 [0, 1, 0, -12136321, 16561802879] [2] 3932160  
73920.er4 73920gd3 [0, 1, 0, -1628481, -422354241] [2] 1966080  
73920.er5 73920gd2 [0, 1, 0, -766081, 253249919] [2, 2] 983040  
73920.er6 73920gd1 [0, 1, 0, 2239, 11843775] [2] 491520 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 73920.er have rank \(1\).

Complex multiplication

The elliptic curves in class 73920.er do not have complex multiplication.

Modular form 73920.2.a.er

sage: E.q_eigenform(10)
 
\( q + q^{3} - q^{5} - q^{7} + q^{9} - q^{11} + 2q^{13} - q^{15} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.