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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 86490.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86490.bi1 | 86490j4 | \([1, -1, 0, -1104369, 446970725]\) | \(8527173507/200\) | \(3493746990624600\) | \([2]\) | \(1451520\) | \(2.0945\) | |
| 86490.bi2 | 86490j3 | \([1, -1, 0, -66489, 7532333]\) | \(-1860867/320\) | \(-5589995184999360\) | \([2]\) | \(725760\) | \(1.7479\) | |
| 86490.bi3 | 86490j2 | \([1, -1, 0, -23244, -350750]\) | \(57960603/31250\) | \(748831230843750\) | \([2]\) | \(483840\) | \(1.5452\) | |
| 86490.bi4 | 86490j1 | \([1, -1, 0, 5586, -45152]\) | \(804357/500\) | \(-11981299693500\) | \([2]\) | \(241920\) | \(1.1986\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 86490.bi have rank \(1\).
Complex multiplication
The elliptic curves in class 86490.bi do not have complex multiplication.Modular form 86490.2.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 LMFDB 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 LMFDB labels.
