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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 405720.bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405720.bo1 | 405720bo3 | \([0, 0, 0, -1518363, 720116838]\) | \(4407931365156/100625\) | \(8837341107840000\) | \([2]\) | \(3932160\) | \(2.1733\) | \(\Gamma_0(N)\)-optimal* |
405720.bo2 | 405720bo4 | \([0, 0, 0, -407043, -89442738]\) | \(84923690436/9794435\) | \(860191434072714240\) | \([2]\) | \(3932160\) | \(2.1733\) | |
405720.bo3 | 405720bo2 | \([0, 0, 0, -98343, 10390842]\) | \(4790692944/648025\) | \(14228119183622400\) | \([2, 2]\) | \(1966080\) | \(1.8268\) | \(\Gamma_0(N)\)-optimal* |
405720.bo4 | 405720bo1 | \([0, 0, 0, 9702, 861273]\) | \(73598976/276115\) | \(-378900999998640\) | \([2]\) | \(983040\) | \(1.4802\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 405720.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 405720.bo do not have complex multiplication.Modular form 405720.2.a.bo
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.