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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 65520.ej
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 65520.ej1 | 65520eo4 | \([0, 0, 0, -19568667, -33318848054]\) | \(277536408914951281369/2063880\) | \(6162712657920\) | \([2]\) | \(1769472\) | \(2.5051\) | |
| 65520.ej2 | 65520eo3 | \([0, 0, 0, -1309467, -442807094]\) | \(83161039719198169/19757817763320\) | \(58996527716189306880\) | \([4]\) | \(1769472\) | \(2.5051\) | |
| 65520.ej3 | 65520eo2 | \([0, 0, 0, -1223067, -520584374]\) | \(67762119444423769/5843073600\) | \(17447324280422400\) | \([2, 2]\) | \(884736\) | \(2.1585\) | |
| 65520.ej4 | 65520eo1 | \([0, 0, 0, -71067, -9326774]\) | \(-13293525831769/4892160000\) | \(-14607911485440000\) | \([2]\) | \(442368\) | \(1.8120\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 65520.ej have rank \(0\).
Complex multiplication
The elliptic curves in class 65520.ej do not have complex multiplication.Modular form 65520.2.a.ej
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
