Properties

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

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E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 1989e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1989.e4 1989e1 \([1, -1, 0, -4851, -128840]\) \(17319700013617/25857\) \(18849753\) \([2]\) \(1024\) \(0.66474\) \(\Gamma_0(N)\)-optimal
1989.e3 1989e2 \([1, -1, 0, -4896, -126293]\) \(17806161424897/668584449\) \(487398063321\) \([2, 2]\) \(2048\) \(1.0113\)  
1989.e2 1989e3 \([1, -1, 0, -12501, 368032]\) \(296380748763217/92608836489\) \(67511841800481\) \([2, 2]\) \(4096\) \(1.3579\)  
1989.e5 1989e4 \([1, -1, 0, 1989, -458150]\) \(1193377118543/124806800313\) \(-90984157428177\) \([2]\) \(4096\) \(1.3579\)  
1989.e1 1989e5 \([1, -1, 0, -181566, 29819155]\) \(908031902324522977/161726530797\) \(117898640951013\) \([2]\) \(8192\) \(1.7045\)  
1989.e6 1989e6 \([1, -1, 0, 34884, 2462449]\) \(6439735268725823/7345472585373\) \(-5354849514736917\) \([2]\) \(8192\) \(1.7045\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 1989e do not have complex multiplication.

Modular form 1989.2.a.e

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.