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SageMath
E = EllipticCurve("gj1")
E.isogeny_class()
Elliptic curves in class 232050gj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.gj4 | 232050gj1 | \([1, 0, 0, -355088, -64958208]\) | \(316892346232279609/66830400000000\) | \(1044225000000000000\) | \([2]\) | \(4423680\) | \(2.1721\) | \(\Gamma_0(N)\)-optimal |
| 232050.gj2 | 232050gj2 | \([1, 0, 0, -5355088, -4769958208]\) | \(1086934883783829079609/69785974440000\) | \(1090405850625000000\) | \([2, 2]\) | \(8847360\) | \(2.5187\) | |
| 232050.gj3 | 232050gj3 | \([1, 0, 0, -5030088, -5374133208]\) | \(-900804278922017287609/277087063526418600\) | \(-4329485367600290625000\) | \([2]\) | \(17694720\) | \(2.8653\) | |
| 232050.gj1 | 232050gj4 | \([1, 0, 0, -85680088, -305265783208]\) | \(4451879473171293653671609/18353298600\) | \(286770290625000\) | \([2]\) | \(17694720\) | \(2.8653\) |
Rank
sage: E.rank()
The elliptic curves in class 232050gj have rank \(0\).
Complex multiplication
The elliptic curves in class 232050gj do not have complex multiplication.Modular form 232050.2.a.gj
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
