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SageMath
E = EllipticCurve("pf1")
E.isogeny_class()
Elliptic curves in class 187200.pf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.pf1 | 187200bx2 | \([0, 0, 0, -131011500, -531840350000]\) | \(666276475992821/58199166792\) | \(21722722606780416000000000\) | \([2]\) | \(47185920\) | \(3.6023\) | |
187200.pf2 | 187200bx1 | \([0, 0, 0, -128131500, -558249950000]\) | \(623295446073461/5458752\) | \(2037468266496000000000\) | \([2]\) | \(23592960\) | \(3.2557\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.pf have rank \(1\).
Complex multiplication
The elliptic curves in class 187200.pf do not have complex multiplication.Modular form 187200.2.a.pf
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.