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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 4800bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4800.r5 | 4800bi1 | \([0, -1, 0, -1533, -22563]\) | \(24918016/45\) | \(720000000\) | \([2]\) | \(3072\) | \(0.59157\) | \(\Gamma_0(N)\)-optimal |
| 4800.r4 | 4800bi2 | \([0, -1, 0, -2033, -6063]\) | \(3631696/2025\) | \(518400000000\) | \([2, 2]\) | \(6144\) | \(0.93814\) | |
| 4800.r2 | 4800bi3 | \([0, -1, 0, -20033, 1091937]\) | \(868327204/5625\) | \(5760000000000\) | \([2, 2]\) | \(12288\) | \(1.2847\) | |
| 4800.r6 | 4800bi4 | \([0, -1, 0, 7967, -56063]\) | \(54607676/32805\) | \(-33592320000000\) | \([2]\) | \(12288\) | \(1.2847\) | |
| 4800.r1 | 4800bi5 | \([0, -1, 0, -320033, 69791937]\) | \(1770025017602/75\) | \(153600000000\) | \([2]\) | \(24576\) | \(1.6313\) | |
| 4800.r3 | 4800bi6 | \([0, -1, 0, -8033, 2375937]\) | \(-27995042/1171875\) | \(-2400000000000000\) | \([2]\) | \(24576\) | \(1.6313\) |
Rank
sage: E.rank()
The elliptic curves in class 4800bi have rank \(0\).
Complex multiplication
The elliptic curves in class 4800bi do not have complex multiplication.Modular form 4800.2.a.bi
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
