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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 489762.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
489762.d1 | 489762d1 | \([1, -1, 0, -3604041, -2918169747]\) | \(-51514785673/6816096\) | \(-685009745131483096416\) | \([]\) | \(29652480\) | \(2.7308\) | \(\Gamma_0(N)\)-optimal* |
489762.d2 | 489762d2 | \([1, -1, 0, 23386104, 7267910976]\) | \(14074607107847/8372453376\) | \(-841421856913251399991296\) | \([]\) | \(88957440\) | \(3.2801\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 489762.d have rank \(1\).
Complex multiplication
The elliptic curves in class 489762.d do not have complex multiplication.Modular form 489762.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.