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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 19890.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19890.bc1 | 19890bd3 | \([1, -1, 1, -12587, 527361]\) | \(302503589987689/12214946250\) | \(8904695816250\) | \([2]\) | \(65536\) | \(1.2509\) | |
| 19890.bc2 | 19890bd2 | \([1, -1, 1, -2057, -24411]\) | \(1319778683209/395612100\) | \(288401220900\) | \([2, 2]\) | \(32768\) | \(0.90431\) | |
| 19890.bc3 | 19890bd1 | \([1, -1, 1, -1877, -30819]\) | \(1002702430729/159120\) | \(115998480\) | \([2]\) | \(16384\) | \(0.55773\) | \(\Gamma_0(N)\)-optimal |
| 19890.bc4 | 19890bd4 | \([1, -1, 1, 5593, -168231]\) | \(26546265663191/31856082570\) | \(-23223084193530\) | \([2]\) | \(65536\) | \(1.2509\) |
Rank
sage: E.rank()
The elliptic curves in class 19890.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 19890.bc do not have complex multiplication.Modular form 19890.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
