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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 67335.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67335.f1 | 67335e2 | \([0, -1, 1, -13472985, 21028830023]\) | \(-2989967081734144/380653171875\) | \(-34433270095310792296875\) | \([]\) | \(6462720\) | \(3.0582\) | |
67335.f2 | 67335e1 | \([0, -1, 1, 1071375, -65218894]\) | \(1503484706816/890163675\) | \(-80522765906111511075\) | \([]\) | \(2154240\) | \(2.5089\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 67335.f have rank \(1\).
Complex multiplication
The elliptic curves in class 67335.f do not have complex multiplication.Modular form 67335.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.