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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 69828r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69828.c2 | 69828r1 | \([0, -1, 0, -40909, -3688286]\) | \(-3196715008/649539\) | \(-1538481332882736\) | \([2]\) | \(337920\) | \(1.6365\) | \(\Gamma_0(N)\)-optimal |
| 69828.c1 | 69828r2 | \([0, -1, 0, -683644, -217333400]\) | \(932410994128/29403\) | \(1114291006532352\) | \([2]\) | \(675840\) | \(1.9831\) |
Rank
sage: E.rank()
The elliptic curves in class 69828r have rank \(0\).
Complex multiplication
The elliptic curves in class 69828r do not have complex multiplication.Modular form 69828.2.a.r
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.
