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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 471510cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
471510.cd2 | 471510cd1 | \([1, -1, 0, 3771, 1301535]\) | \(1685159/209250\) | \(-736297131989250\) | \([]\) | \(2211840\) | \(1.5321\) | \(\Gamma_0(N)\)-optimal* |
471510.cd1 | 471510cd2 | \([1, -1, 0, -794754, 272959740]\) | \(-15777367606441/3574920\) | \(-12579227446074120\) | \([]\) | \(6635520\) | \(2.0814\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 471510cd have rank \(1\).
Complex multiplication
The elliptic curves in class 471510cd do not have complex multiplication.Modular form 471510.2.a.cd
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 Cremona 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 Cremona labels.