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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 66240do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66240.cn2 | 66240do1 | \([0, 0, 0, 132, -8208]\) | \(574992/66125\) | \(-29251584000\) | \([2]\) | \(36864\) | \(0.68765\) | \(\Gamma_0(N)\)-optimal |
66240.cn1 | 66240do2 | \([0, 0, 0, -5388, -147312]\) | \(9776035692/359375\) | \(635904000000\) | \([2]\) | \(73728\) | \(1.0342\) |
Rank
sage: E.rank()
The elliptic curves in class 66240do have rank \(1\).
Complex multiplication
The elliptic curves in class 66240do do not have complex multiplication.Modular form 66240.2.a.do
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.