Properties

Label 1989e
Number of curves $6$
Conductor $1989$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1989.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1989e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1989.e4 1989e1 [1, -1, 0, -4851, -128840] [2] 1024 \(\Gamma_0(N)\)-optimal
1989.e3 1989e2 [1, -1, 0, -4896, -126293] [2, 2] 2048  
1989.e2 1989e3 [1, -1, 0, -12501, 368032] [2, 2] 4096  
1989.e5 1989e4 [1, -1, 0, 1989, -458150] [2] 4096  
1989.e1 1989e5 [1, -1, 0, -181566, 29819155] [2] 8192  
1989.e6 1989e6 [1, -1, 0, 34884, 2462449] [2] 8192  

Rank

sage: E.rank()
 

The elliptic curves in class 1989e have rank \(1\).

Modular form 1989.2.a.e

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} + 2q^{5} - 3q^{8} + 2q^{10} - 4q^{11} + q^{13} - q^{16} - q^{17} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.