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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 195730a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 195730.k3 | 195730a1 | \([1, 0, 0, -39686, 1660516]\) | \(46694890801/18944000\) | \(2804391881216000\) | \([2]\) | \(1216512\) | \(1.6619\) | \(\Gamma_0(N)\)-optimal |
| 195730.k4 | 195730a2 | \([1, 0, 0, 129594, 12122020]\) | \(1625964918479/1369000000\) | \(-202661132041000000\) | \([2]\) | \(2433024\) | \(2.0085\) | |
| 195730.k1 | 195730a3 | \([1, 0, 0, -2790486, 1793954836]\) | \(16232905099479601/4052240\) | \(599876950841360\) | \([2]\) | \(3649536\) | \(2.2113\) | |
| 195730.k2 | 195730a4 | \([1, 0, 0, -2779906, 1808235720]\) | \(-16048965315233521/256572640900\) | \(-37981958988709260100\) | \([2]\) | \(7299072\) | \(2.5578\) |
Rank
sage: E.rank()
The elliptic curves in class 195730a have rank \(1\).
Complex multiplication
The elliptic curves in class 195730a do not have complex multiplication.Modular form 195730.2.a.a
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.
