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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 6864.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6864.o1 | 6864o1 | \([0, -1, 0, -1663784, -825471888]\) | \(-124352595912593543977/103332962304\) | \(-423251813597184\) | \([]\) | \(74880\) | \(2.1099\) | \(\Gamma_0(N)\)-optimal |
| 6864.o2 | 6864o2 | \([0, -1, 0, -1291544, -1204934928]\) | \(-58169016237585194137/119573538788081664\) | \(-489773214875982495744\) | \([]\) | \(224640\) | \(2.6592\) |
Rank
sage: E.rank()
The elliptic curves in class 6864.o have rank \(1\).
Complex multiplication
The elliptic curves in class 6864.o do not have complex multiplication.Modular form 6864.2.a.o
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
