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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 253575eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 253575.eg4 | 253575eg1 | \([1, -1, 0, 5032683, 2734452216]\) | \(10519294081031/8500170375\) | \(-11391041264107271484375\) | \([2]\) | \(16588800\) | \(2.9201\) | \(\Gamma_0(N)\)-optimal |
| 253575.eg3 | 253575eg2 | \([1, -1, 0, -24128442, 23817945591]\) | \(1159246431432649/488076890625\) | \(654069713416367431640625\) | \([2, 2]\) | \(33177600\) | \(3.2667\) | |
| 253575.eg1 | 253575eg3 | \([1, -1, 0, -332222067, 2329898728716]\) | \(3026030815665395929/1364501953125\) | \(1828563119007110595703125\) | \([2]\) | \(66355200\) | \(3.6133\) | |
| 253575.eg2 | 253575eg4 | \([1, -1, 0, -182612817, -933269195034]\) | \(502552788401502649/10024505152875\) | \(13433795654790636685546875\) | \([2]\) | \(66355200\) | \(3.6133\) |
Rank
sage: E.rank()
The elliptic curves in class 253575eg have rank \(1\).
Complex multiplication
The elliptic curves in class 253575eg do not have complex multiplication.Modular form 253575.2.a.eg
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
