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SageMath
E = EllipticCurve("dz1")
E.isogeny_class()
Elliptic curves in class 123840.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
123840.dz1 | 123840ec2 | \([0, 0, 0, -6372, -195264]\) | \(354894912/1075\) | \(86668185600\) | \([2]\) | \(147456\) | \(0.96760\) | |
123840.dz2 | 123840ec1 | \([0, 0, 0, -567, -216]\) | \(16003008/9245\) | \(11646037440\) | \([2]\) | \(73728\) | \(0.62103\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 123840.dz have rank \(1\).
Complex multiplication
The elliptic curves in class 123840.dz do not have complex multiplication.Modular form 123840.2.a.dz
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.