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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 89232.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
89232.f1 | 89232c2 | \([0, -1, 0, -7995784, -7146677552]\) | \(5718957389087906/1075876263891\) | \(10635364830075809421312\) | \([2]\) | \(6322176\) | \(2.9444\) | |
89232.f2 | 89232c1 | \([0, -1, 0, 1001776, -654038256]\) | \(22494434350748/50367250791\) | \(-248947813809414061056\) | \([2]\) | \(3161088\) | \(2.5978\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 89232.f have rank \(1\).
Complex multiplication
The elliptic curves in class 89232.f do not have complex multiplication.Modular form 89232.2.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 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.