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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 72450.do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.do1 | 72450cu2 | \([1, -1, 1, -169265, -26761563]\) | \(158909194494247023/5080516\) | \(17146741500\) | \([2]\) | \(294912\) | \(1.4662\) | |
72450.do2 | 72450cu1 | \([1, -1, 1, -10565, -417363]\) | \(-38638468208943/219395344\) | \(-740459286000\) | \([2]\) | \(147456\) | \(1.1196\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.do have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.do do not have complex multiplication.Modular form 72450.2.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 LMFDB 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 LMFDB labels.