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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 216384.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
216384.by1 | 216384df2 | \([0, -1, 0, -1130593, 455976193]\) | \(5182207647625/91449288\) | \(2820385612473827328\) | \([2]\) | \(3538944\) | \(2.3365\) | |
216384.by2 | 216384df1 | \([0, -1, 0, -1633, 20423425]\) | \(-15625/5842368\) | \(-180184351462391808\) | \([2]\) | \(1769472\) | \(1.9899\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 216384.by have rank \(1\).
Complex multiplication
The elliptic curves in class 216384.by do not have complex multiplication.Modular form 216384.2.a.by
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.