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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 174240.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 174240.cm1 | 174240cg4 | \([0, 0, 0, -3921852, -2989404704]\) | \(1261112198464/675\) | \(3570650640691200\) | \([2]\) | \(3932160\) | \(2.3139\) | |
| 174240.cm2 | 174240cg2 | \([0, 0, 0, -540507, 85149394]\) | \(26410345352/10546875\) | \(6973927032600000000\) | \([2]\) | \(3932160\) | \(2.3139\) | |
| 174240.cm3 | 174240cg1 | \([0, 0, 0, -246477, -46164404]\) | \(20034997696/455625\) | \(37659205976040000\) | \([2, 2]\) | \(1966080\) | \(1.9673\) | \(\Gamma_0(N)\)-optimal |
| 174240.cm4 | 174240cg3 | \([0, 0, 0, 25773, -142595354]\) | \(2863288/13286025\) | \(-8785139570090611200\) | \([2]\) | \(3932160\) | \(2.3139\) |
Rank
sage: E.rank()
The elliptic curves in class 174240.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 174240.cm do not have complex multiplication.Modular form 174240.2.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 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.
