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SageMath
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)
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.