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SageMath
E = EllipticCurve("ij1")
E.isogeny_class()
Elliptic curves in class 463680.ij
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.ij1 | 463680ij2 | \([0, 0, 0, -5286252, 1091610704]\) | \(85486955243540761/46777901234400\) | \(8939396136927913574400\) | \([2]\) | \(19660800\) | \(2.9028\) | \(\Gamma_0(N)\)-optimal* |
463680.ij2 | 463680ij1 | \([0, 0, 0, -3166572, -2154891184]\) | \(18374873741826841/136564270080\) | \(26097838427555758080\) | \([2]\) | \(9830400\) | \(2.5562\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.ij have rank \(0\).
Complex multiplication
The elliptic curves in class 463680.ij do not have complex multiplication.Modular form 463680.2.a.ij
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.