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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 417450s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
417450.s2 | 417450s1 | \([1, 1, 0, -6479310, -19018287900]\) | \(-135845097606008981/626788691174448\) | \(-138799300065712034166000\) | \([2]\) | \(61931520\) | \(3.1253\) | \(\Gamma_0(N)\)-optimal* |
417450.s1 | 417450s2 | \([1, 1, 0, -153700010, -732302579400]\) | \(1813340159520283008341/3408095894210172\) | \(754706221305358314811500\) | \([2]\) | \(123863040\) | \(3.4719\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 417450s have rank \(1\).
Complex multiplication
The elliptic curves in class 417450s do not have complex multiplication.Modular form 417450.2.a.s
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.