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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 190608bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
190608.v2 | 190608bt1 | \([0, -1, 0, 45727, 38506680]\) | \(38912000/2381643\) | \(-647180544506501808\) | \([]\) | \(1477440\) | \(2.0972\) | \(\Gamma_0(N)\)-optimal |
190608.v1 | 190608bt2 | \([0, -1, 0, -7361993, 7692903756]\) | \(-162390710272000/47832147\) | \(-12997764543374064432\) | \([]\) | \(4432320\) | \(2.6465\) |
Rank
sage: E.rank()
The elliptic curves in class 190608bt have rank \(1\).
Complex multiplication
The elliptic curves in class 190608bt do not have complex multiplication.Modular form 190608.2.a.bt
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.