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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 85291a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85291.a3 | 85291a1 | \([0, -1, 1, 2993, 1787]\) | \(32768/19\) | \(-1718709261211\) | \([]\) | \(101376\) | \(1.0372\) | \(\Gamma_0(N)\)-optimal |
| 85291.a2 | 85291a2 | \([0, -1, 1, -41897, 3525652]\) | \(-89915392/6859\) | \(-620454043297171\) | \([]\) | \(304128\) | \(1.5865\) | |
| 85291.a1 | 85291a3 | \([0, -1, 1, -3453537, 2471420737]\) | \(-50357871050752/19\) | \(-1718709261211\) | \([]\) | \(912384\) | \(2.1358\) |
Rank
sage: E.rank()
The elliptic curves in class 85291a have rank \(1\).
Complex multiplication
The elliptic curves in class 85291a do not have complex multiplication.Modular form 85291.2.a.a
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.
