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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 106134.co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106134.co1 | 106134df2 | \([1, 0, 0, -12586092, -17275185264]\) | \(-16591834777/98304\) | \(-1306391047362907570176\) | \([]\) | \(8709120\) | \(2.8921\) | |
106134.co2 | 106134df1 | \([1, 0, 0, 415323, -126318879]\) | \(596183/864\) | \(-11481952564713054816\) | \([]\) | \(2903040\) | \(2.3428\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 106134.co have rank \(1\).
Complex multiplication
The elliptic curves in class 106134.co do not have complex multiplication.Modular form 106134.2.a.co
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.