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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1989.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1989.d1 | 1989a2 | \([1, -1, 0, -882, -7533]\) | \(104154702625/24649677\) | \(17969614533\) | \([2]\) | \(1024\) | \(0.67920\) | |
| 1989.d2 | 1989a1 | \([1, -1, 0, -297, 1944]\) | \(3981876625/232713\) | \(169647777\) | \([2]\) | \(512\) | \(0.33263\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1989.d have rank \(0\).
Complex multiplication
The elliptic curves in class 1989.d do not have complex multiplication.Modular form 1989.2.a.d
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.
