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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 6630.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6630.j1 | 6630j3 | \([1, 0, 1, -307684, -65712898]\) | \(3221338935539503699129/200350631681460\) | \(200350631681460\) | \([2]\) | \(51200\) | \(1.8036\) | |
| 6630.j2 | 6630j4 | \([1, 0, 1, -103004, 11924606]\) | \(120859257477573578809/8424459021127500\) | \(8424459021127500\) | \([2]\) | \(51200\) | \(1.8036\) | |
| 6630.j3 | 6630j2 | \([1, 0, 1, -20384, -898018]\) | \(936615448738871929/194959225328400\) | \(194959225328400\) | \([2, 2]\) | \(25600\) | \(1.4570\) | |
| 6630.j4 | 6630j1 | \([1, 0, 1, 2736, -84194]\) | \(2266209994236551/4390344840960\) | \(-4390344840960\) | \([2]\) | \(12800\) | \(1.1104\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6630.j have rank \(1\).
Complex multiplication
The elliptic curves in class 6630.j do not have complex multiplication.Modular form 6630.2.a.j
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.
