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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 188760.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 188760.bi1 | 188760ce5 | \([0, -1, 0, -15691320, 23819118732]\) | \(117764966889591842/626999726925\) | \(2274853403097046886400\) | \([2]\) | \(9830400\) | \(2.9425\) | |
| 188760.bi2 | 188760ce3 | \([0, -1, 0, -1534320, -94885668]\) | \(220199214811684/125262905625\) | \(227236739432376960000\) | \([2, 2]\) | \(4915200\) | \(2.5959\) | |
| 188760.bi3 | 188760ce2 | \([0, -1, 0, -1125340, -458223500]\) | \(347519589019216/777573225\) | \(352644710413881600\) | \([2, 2]\) | \(2457600\) | \(2.2493\) | |
| 188760.bi4 | 188760ce1 | \([0, -1, 0, -1124735, -458742348]\) | \(5551350318708736/27885\) | \(790399655760\) | \([2]\) | \(1228800\) | \(1.9028\) | \(\Gamma_0(N)\)-optimal |
| 188760.bi5 | 188760ce4 | \([0, -1, 0, -726040, -788364740]\) | \(-23331888216004/134595568965\) | \(-244166923009233269760\) | \([2]\) | \(4915200\) | \(2.5959\) | |
| 188760.bi6 | 188760ce6 | \([0, -1, 0, 6079000, -761812500]\) | \(6847531008289918/4031426953125\) | \(-14626649629706400000000\) | \([2]\) | \(9830400\) | \(2.9425\) |
Rank
sage: E.rank()
The elliptic curves in class 188760.bi have rank \(1\).
Complex multiplication
The elliptic curves in class 188760.bi do not have complex multiplication.Modular form 188760.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
