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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 242550be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.be4 | 242550be1 | \([1, -1, 0, 2827683, 2435950341]\) | \(1865864036231/2993760000\) | \(-4011924725077500000000\) | \([2]\) | \(11796480\) | \(2.8305\) | \(\Gamma_0(N)\)-optimal |
242550.be3 | 242550be2 | \([1, -1, 0, -19222317, 24860800341]\) | \(586145095611769/140040608400\) | \(187667808827312756250000\) | \([2, 2]\) | \(23592960\) | \(3.1770\) | |
242550.be1 | 242550be3 | \([1, -1, 0, -287129817, 1872618827841]\) | \(1953542217204454969/170843779260\) | \(228947003814225788437500\) | \([2]\) | \(47185920\) | \(3.5236\) | |
242550.be2 | 242550be4 | \([1, -1, 0, -104114817, -387971427159]\) | \(93137706732176569/5369647977540\) | \(7195841846392201911562500\) | \([2]\) | \(47185920\) | \(3.5236\) |
Rank
sage: E.rank()
The elliptic curves in class 242550be have rank \(1\).
Complex multiplication
The elliptic curves in class 242550be do not have complex multiplication.Modular form 242550.2.a.be
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.