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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 6630.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6630.k1 | 6630k3 | \([1, 0, 1, -166134, -26077304]\) | \(507102228823216499929/2648775168000\) | \(2648775168000\) | \([2]\) | \(34560\) | \(1.5799\) | |
| 6630.k2 | 6630k4 | \([1, 0, 1, -163254, -27024248]\) | \(-481184224995688814809/36713242449000000\) | \(-36713242449000000\) | \([2]\) | \(69120\) | \(1.9265\) | |
| 6630.k3 | 6630k1 | \([1, 0, 1, -2919, -2918]\) | \(2749236527524969/1587903192720\) | \(1587903192720\) | \([6]\) | \(11520\) | \(1.0306\) | \(\Gamma_0(N)\)-optimal |
| 6630.k4 | 6630k2 | \([1, 0, 1, 11661, -20414]\) | \(175381844946241751/101691694692900\) | \(-101691694692900\) | \([6]\) | \(23040\) | \(1.3772\) |
Rank
sage: E.rank()
The elliptic curves in class 6630.k have rank \(1\).
Complex multiplication
The elliptic curves in class 6630.k do not have complex multiplication.Modular form 6630.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
