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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 242550gv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242550.gv4 | 242550gv1 | \([1, -1, 0, 321333, -323651259]\) | \(73929353373/954060800\) | \(-47353063665600000000\) | \([2]\) | \(7962624\) | \(2.4562\) | \(\Gamma_0(N)\)-optimal |
| 242550.gv2 | 242550gv2 | \([1, -1, 0, -5558667, -4716011259]\) | \(382704614800227/27778076480\) | \(1378714044288735000000\) | \([2]\) | \(15925248\) | \(2.8027\) | |
| 242550.gv3 | 242550gv3 | \([1, -1, 0, -2912667, 9092678741]\) | \(-75526045083/943250000\) | \(-34129220751527343750000\) | \([2]\) | \(23887872\) | \(3.0055\) | |
| 242550.gv1 | 242550gv4 | \([1, -1, 0, -85600167, 303873616241]\) | \(1917114236485083/7117764500\) | \(257539099791025335937500\) | \([2]\) | \(47775744\) | \(3.3520\) |
Rank
sage: E.rank()
The elliptic curves in class 242550gv have rank \(1\).
Complex multiplication
The elliptic curves in class 242550gv do not have complex multiplication.Modular form 242550.2.a.gv
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 Cremona 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 Cremona labels.
