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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 75810.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
75810.df1 | 75810dc1 | \([1, 0, 0, -6591, -206379]\) | \(4616586342451/3307500\) | \(22686142500\) | \([2]\) | \(138240\) | \(0.92238\) | \(\Gamma_0(N)\)-optimal |
75810.df2 | 75810dc2 | \([1, 0, 0, -5261, -291765]\) | \(-2347864201171/3986718750\) | \(-27344903906250\) | \([2]\) | \(276480\) | \(1.2690\) |
Rank
sage: E.rank()
The elliptic curves in class 75810.df have rank \(0\).
Complex multiplication
The elliptic curves in class 75810.df do not have complex multiplication.Modular form 75810.2.a.df
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.