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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 408629f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
408629.f2 | 408629f1 | \([1, -1, 0, -1354739, -605591616]\) | \(43499078731809/82055753\) | \(518704204972070897\) | \([2]\) | \(9676800\) | \(2.2892\) | \(\Gamma_0(N)\)-optimal |
408629.f1 | 408629f2 | \([1, -1, 0, -21666004, -38811081081]\) | \(177930109857804849/634933\) | \(4013642004790717\) | \([2]\) | \(19353600\) | \(2.6358\) |
Rank
sage: E.rank()
The elliptic curves in class 408629f have rank \(1\).
Complex multiplication
The elliptic curves in class 408629f do not have complex multiplication.Modular form 408629.2.a.f
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.