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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 20286bt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20286.bi2 | 20286bt1 | \([1, -1, 1, -55649, 6800049]\) | \(-5999796014211/2790817792\) | \(-8865096905097216\) | \([]\) | \(253440\) | \(1.7663\) | \(\Gamma_0(N)\)-optimal |
20286.bi1 | 20286bt2 | \([1, -1, 1, -4924289, 4207174129]\) | \(-5702623460245179/252448\) | \(-584590114283616\) | \([]\) | \(760320\) | \(2.3156\) |
Rank
sage: E.rank()
The elliptic curves in class 20286bt have rank \(2\).
Complex multiplication
The elliptic curves in class 20286bt do not have complex multiplication.Modular form 20286.2.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.