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SageMath
E = EllipticCurve("df1")
E.isogeny_class()
Elliptic curves in class 476850.df
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.df1 | 476850df1 | \([1, 0, 1, -6531551, 6373664498]\) | \(81706955619457/744505344\) | \(280789829869824000000\) | \([2]\) | \(41287680\) | \(2.7459\) | \(\Gamma_0(N)\)-optimal |
476850.df2 | 476850df2 | \([1, 0, 1, -1907551, 15224000498]\) | \(-2035346265217/264305213568\) | \(-99682583274333378000000\) | \([2]\) | \(82575360\) | \(3.0925\) |
Rank
sage: E.rank()
The elliptic curves in class 476850.df have rank \(1\).
Complex multiplication
The elliptic curves in class 476850.df do not have complex multiplication.Modular form 476850.2.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.