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SageMath
E = EllipticCurve("ce1")
E.isogeny_class()
Elliptic curves in class 106722ce
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.bf2 | 106722ce1 | \([1, -1, 0, -32643, 7511125]\) | \(-33698267/193536\) | \(-22093045383822336\) | \([2]\) | \(1105920\) | \(1.8201\) | \(\Gamma_0(N)\)-optimal |
| 106722.bf1 | 106722ce2 | \([1, -1, 0, -808803, 279632821]\) | \(512576216027/1143072\) | \(130487049298200672\) | \([2]\) | \(2211840\) | \(2.1666\) |
Rank
sage: E.rank()
The elliptic curves in class 106722ce have rank \(1\).
Complex multiplication
The elliptic curves in class 106722ce do not have complex multiplication.Modular form 106722.2.a.ce
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
