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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 61710cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 61710.cl3 | 61710cm1 | \([1, 0, 0, -16761, 698985]\) | \(293946977449/50490000\) | \(89446114890000\) | \([2]\) | \(276480\) | \(1.3974\) | \(\Gamma_0(N)\)-optimal |
| 61710.cl2 | 61710cm2 | \([1, 0, 0, -77261, -7613715]\) | \(28790481449449/2549240100\) | \(4516134340796100\) | \([2, 2]\) | \(552960\) | \(1.7440\) | |
| 61710.cl4 | 61710cm3 | \([1, 0, 0, 86089, -35481225]\) | \(39829997144951/330164359470\) | \(-584906302827032670\) | \([2]\) | \(1105920\) | \(2.0906\) | |
| 61710.cl1 | 61710cm4 | \([1, 0, 0, -1208611, -511517005]\) | \(110211585818155849/993794670\) | \(1760567879379870\) | \([2]\) | \(1105920\) | \(2.0906\) |
Rank
sage: E.rank()
The elliptic curves in class 61710cm have rank \(1\).
Complex multiplication
The elliptic curves in class 61710cm do not have complex multiplication.Modular form 61710.2.a.cm
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.
