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SageMath
E = EllipticCurve("p1")
E.isogeny_class()
Elliptic curves in class 106722p
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.cd4 | 106722p1 | \([1, -1, 0, -383532, 88249552]\) | \(1108717875/45056\) | \(253548326446829568\) | \([2]\) | \(1382400\) | \(2.1054\) | \(\Gamma_0(N)\)-optimal |
| 106722.cd2 | 106722p2 | \([1, -1, 0, -6075372, 5765290768]\) | \(4406910829875/7744\) | \(43578618608048832\) | \([2]\) | \(2764800\) | \(2.4520\) | |
| 106722.cd3 | 106722p3 | \([1, -1, 0, -4652412, -3835040896]\) | \(2714704875/21296\) | \(87364235654485895952\) | \([2]\) | \(4147200\) | \(2.6547\) | |
| 106722.cd1 | 106722p4 | \([1, -1, 0, -7854072, 2119406372]\) | \(13060888875/7086244\) | \(29070449414030181878028\) | \([2]\) | \(8294400\) | \(3.0013\) |
Rank
sage: E.rank()
The elliptic curves in class 106722p have rank \(1\).
Complex multiplication
The elliptic curves in class 106722p do not have complex multiplication.Modular form 106722.2.a.p
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
