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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 72450.bi
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.bi1 | 72450bv4 | \([1, -1, 0, -189292796442, 31699316105513716]\) | \(65853432878493908038433301506521/38511703125000000\) | \(438672368408203125000000\) | \([2]\) | \(247726080\) | \(4.7618\) | |
| 72450.bi2 | 72450bv2 | \([1, -1, 0, -11830868442, 495297995633716]\) | \(16077778198622525072705635801/388799208512064000000\) | \(4428665984457729000000000000\) | \([2, 2]\) | \(123863040\) | \(4.4152\) | |
| 72450.bi3 | 72450bv3 | \([1, -1, 0, -11389868442, 533924744633716]\) | \(-14346048055032350809895395801/2509530875136386550792000\) | \(-28585125124600403055115125000000\) | \([2]\) | \(247726080\) | \(4.7618\) | |
| 72450.bi4 | 72450bv1 | \([1, -1, 0, -767060442, 7129595249716]\) | \(4381924769947287308715481/608122186185572352000\) | \(6926891777020035072000000000\) | \([2]\) | \(61931520\) | \(4.0686\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 72450.bi do not have complex multiplication.Modular form 72450.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
