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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 240669bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
240669.bd2 | 240669bd1 | \([1, -1, 0, -285885, -58710272]\) | \(2000852317801/2094417\) | \(2704872469229073\) | \([2]\) | \(2488320\) | \(1.8784\) | \(\Gamma_0(N)\)-optimal |
240669.bd1 | 240669bd2 | \([1, -1, 0, -356670, -27352517]\) | \(3885442650361/1996623837\) | \(2578575731627377053\) | \([2]\) | \(4976640\) | \(2.2250\) |
Rank
sage: E.rank()
The elliptic curves in class 240669bd have rank \(1\).
Complex multiplication
The elliptic curves in class 240669bd do not have complex multiplication.Modular form 240669.2.a.bd
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.