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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 570.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 570.d1 | 570d3 | \([1, 0, 1, -1233444, -527363678]\) | \(207530301091125281552569/805586668007040\) | \(805586668007040\) | \([2]\) | \(8960\) | \(2.0734\) | |
| 570.d2 | 570d4 | \([1, 0, 1, -233764, 33569186]\) | \(1412712966892699019449/330160465517040000\) | \(330160465517040000\) | \([2]\) | \(8960\) | \(2.0734\) | |
| 570.d3 | 570d2 | \([1, 0, 1, -78244, -7985758]\) | \(52974743974734147769/3152005008998400\) | \(3152005008998400\) | \([2, 2]\) | \(4480\) | \(1.7268\) | |
| 570.d4 | 570d1 | \([1, 0, 1, 3676, -514654]\) | \(5495662324535111/117739817533440\) | \(-117739817533440\) | \([2]\) | \(2240\) | \(1.3803\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 570.d have rank \(0\).
Complex multiplication
The elliptic curves in class 570.d do not have complex multiplication.Modular form 570.2.a.d
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.
