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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 36822bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36822.v1 | 36822bc1 | \([1, 0, 0, -910, -6784]\) | \(1771561/612\) | \(28792079172\) | \([2]\) | \(55296\) | \(0.70869\) | \(\Gamma_0(N)\)-optimal |
36822.v2 | 36822bc2 | \([1, 0, 0, 2700, -46494]\) | \(46268279/46818\) | \(-2202594056658\) | \([2]\) | \(110592\) | \(1.0553\) |
Rank
sage: E.rank()
The elliptic curves in class 36822bc have rank \(0\).
Complex multiplication
The elliptic curves in class 36822bc do not have complex multiplication.Modular form 36822.2.a.bc
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.