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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 166410cm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166410.x2 | 166410cm1 | \([1, -1, 0, -636060969, -6174099116467]\) | \(2097201152384001/64000000\) | \(868480033426864704000000\) | \([2]\) | \(61028352\) | \(3.6911\) | \(\Gamma_0(N)\)-optimal |
166410.x1 | 166410cm2 | \([1, -1, 0, -10176900969, -395156054084467]\) | \(8589947174918144001/8000\) | \(108560004178358088000\) | \([2]\) | \(122056704\) | \(4.0376\) |
Rank
sage: E.rank()
The elliptic curves in class 166410cm have rank \(0\).
Complex multiplication
The elliptic curves in class 166410cm do not have complex multiplication.Modular form 166410.2.a.cm
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.