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SageMath
E = EllipticCurve("gr1")
E.isogeny_class()
Elliptic curves in class 242550.gr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242550.gr1 | 242550gr3 | \([1, -1, 0, -2933890542, 61167281887116]\) | \(2084105208962185000201/31185000\) | \(41790882552890625000\) | \([2]\) | \(113246208\) | \(3.7685\) | |
| 242550.gr2 | 242550gr4 | \([1, -1, 0, -198808542, 785358145116]\) | \(648474704552553481/176469171805080\) | \(236485567840691870229375000\) | \([2]\) | \(113246208\) | \(3.7685\) | |
| 242550.gr3 | 242550gr2 | \([1, -1, 0, -183373542, 955714240116]\) | \(508859562767519881/62240270400\) | \(83407915034361225000000\) | \([2, 2]\) | \(56623104\) | \(3.4219\) | |
| 242550.gr4 | 242550gr1 | \([1, -1, 0, -10501542, 17537896116]\) | \(-95575628340361/43812679680\) | \(-58713181043267520000000\) | \([2]\) | \(28311552\) | \(3.0754\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 242550.gr have rank \(0\).
Complex multiplication
The elliptic curves in class 242550.gr do not have complex multiplication.Modular form 242550.2.a.gr
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.
