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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 286650.dz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.dz1 | 286650dz4 | \([1, -1, 0, -44091042, 112697966366]\) | \(261984288445803/42250\) | \(1528714102042968750\) | \([2]\) | \(23887872\) | \(2.8915\) | |
| 286650.dz2 | 286650dz3 | \([1, -1, 0, -2747292, 1772685116]\) | \(-63378025803/812500\) | \(-29398348116210937500\) | \([2]\) | \(11943936\) | \(2.5449\) | |
| 286650.dz3 | 286650dz2 | \([1, -1, 0, -615792, 111560616]\) | \(520300455507/193072360\) | \(9582793628191875000\) | \([2]\) | \(7962624\) | \(2.3422\) | |
| 286650.dz4 | 286650dz1 | \([1, -1, 0, 119208, 12335616]\) | \(3774555693/3515200\) | \(-174470525775000000\) | \([2]\) | \(3981312\) | \(1.9956\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286650.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 286650.dz do not have complex multiplication.Modular form 286650.2.a.dz
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.
