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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 76050.do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.do1 | 76050dp4 | \([1, -1, 1, -152069105, -721749781853]\) | \(261984288445803/42250\) | \(62718858521261718750\) | \([2]\) | \(13934592\) | \(3.2010\) | |
| 76050.do2 | 76050dp3 | \([1, -1, 1, -9475355, -11347719353]\) | \(-63378025803/812500\) | \(-1206131894639648437500\) | \([2]\) | \(6967296\) | \(2.8544\) | |
| 76050.do3 | 76050dp2 | \([1, -1, 1, -2123855, -713056353]\) | \(520300455507/193072360\) | \(393155186441866875000\) | \([2]\) | \(4644864\) | \(2.6517\) | |
| 76050.do4 | 76050dp1 | \([1, -1, 1, 411145, -79306353]\) | \(3774555693/3515200\) | \(-7158037076775000000\) | \([2]\) | \(2322432\) | \(2.3051\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76050.do have rank \(0\).
Complex multiplication
The elliptic curves in class 76050.do do not have complex multiplication.Modular form 76050.2.a.do
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.
