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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 64715e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64715.c2 | 64715e1 | \([0, -1, 1, -2465, -51447]\) | \(-262144/35\) | \(-221247706715\) | \([]\) | \(51408\) | \(0.90945\) | \(\Gamma_0(N)\)-optimal |
| 64715.c3 | 64715e2 | \([0, -1, 1, 16025, 127906]\) | \(71991296/42875\) | \(-271028440725875\) | \([]\) | \(154224\) | \(1.4588\) | |
| 64715.c1 | 64715e3 | \([0, -1, 1, -242835, 48262923]\) | \(-250523582464/13671875\) | \(-86424885435546875\) | \([]\) | \(462672\) | \(2.0081\) |
Rank
sage: E.rank()
The elliptic curves in class 64715e have rank \(1\).
Complex multiplication
The elliptic curves in class 64715e do not have complex multiplication.Modular form 64715.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
