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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 2550.c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2550.c1 | 2550b5 | \([1, 1, 0, -693600, -222626250]\) | \(2361739090258884097/5202\) | \(81281250\) | \([2]\) | \(16384\) | \(1.6518\) | |
| 2550.c2 | 2550b3 | \([1, 1, 0, -43350, -3492000]\) | \(576615941610337/27060804\) | \(422825062500\) | \([2, 2]\) | \(8192\) | \(1.3052\) | |
| 2550.c3 | 2550b6 | \([1, 1, 0, -41100, -3867750]\) | \(-491411892194497/125563633938\) | \(-1961931780281250\) | \([2]\) | \(16384\) | \(1.6518\) | |
| 2550.c4 | 2550b2 | \([1, 1, 0, -2850, -49500]\) | \(163936758817/30338064\) | \(474032250000\) | \([2, 2]\) | \(4096\) | \(0.95865\) | |
| 2550.c5 | 2550b1 | \([1, 1, 0, -850, 8500]\) | \(4354703137/352512\) | \(5508000000\) | \([2]\) | \(2048\) | \(0.61208\) | \(\Gamma_0(N)\)-optimal |
| 2550.c6 | 2550b4 | \([1, 1, 0, 5650, -279000]\) | \(1276229915423/2927177028\) | \(-45737141062500\) | \([2]\) | \(8192\) | \(1.3052\) |
Rank
sage: E.rank()
The elliptic curves in class 2550.c have rank \(1\).
Complex multiplication
The elliptic curves in class 2550.c do not have complex multiplication.Modular form 2550.2.a.c
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
