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SageMath
E = EllipticCurve("hb1")
E.isogeny_class()
Elliptic curves in class 106722hb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.gk4 | 106722hb1 | \([1, -1, 1, -107834, -215575]\) | \(912673/528\) | \(80224275164817168\) | \([2]\) | \(1105920\) | \(1.9331\) | \(\Gamma_0(N)\)-optimal |
| 106722.gk2 | 106722hb2 | \([1, -1, 1, -1175054, 488998073]\) | \(1180932193/4356\) | \(661850270109741636\) | \([2, 2]\) | \(2211840\) | \(2.2797\) | |
| 106722.gk3 | 106722hb3 | \([1, -1, 1, -641444, 934669145]\) | \(-192100033/2371842\) | \(-360377472074754320802\) | \([2]\) | \(4423680\) | \(2.6263\) | |
| 106722.gk1 | 106722hb4 | \([1, -1, 1, -18784184, 31340193833]\) | \(4824238966273/66\) | \(10028034395602146\) | \([2]\) | \(4423680\) | \(2.6263\) |
Rank
sage: E.rank()
The elliptic curves in class 106722hb have rank \(1\).
Complex multiplication
The elliptic curves in class 106722hb do not have complex multiplication.Modular form 106722.2.a.hb
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.
