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SageMath
E = EllipticCurve("gf1")
E.isogeny_class()
Elliptic curves in class 485100.gf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
485100.gf1 | 485100gf1 | \([0, 0, 0, -7408800, 5048402625]\) | \(226492416/75625\) | \(15016857130672031250000\) | \([2]\) | \(30965760\) | \(2.9579\) | \(\Gamma_0(N)\)-optimal |
485100.gf2 | 485100gf2 | \([0, 0, 0, 21531825, 34828305750]\) | \(347482224/366025\) | \(-1162905416199242100000000\) | \([2]\) | \(61931520\) | \(3.3045\) |
Rank
sage: E.rank()
The elliptic curves in class 485100.gf have rank \(0\).
Complex multiplication
The elliptic curves in class 485100.gf do not have complex multiplication.Modular form 485100.2.a.gf
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.