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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 82110.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 82110.o1 | 82110q4 | \([1, 1, 0, -5686712, -5222003136]\) | \(20337966805091038553406601/24168591095040\) | \(24168591095040\) | \([2]\) | \(2031616\) | \(2.2809\) | |
| 82110.o2 | 82110q2 | \([1, 1, 0, -355512, -81660096]\) | \(4969222174347906673801/5412627298713600\) | \(5412627298713600\) | \([2, 2]\) | \(1015808\) | \(1.9344\) | |
| 82110.o3 | 82110q3 | \([1, 1, 0, -267192, -123117504]\) | \(-2109582937351555472521/5315560665303840000\) | \(-5315560665303840000\) | \([4]\) | \(2031616\) | \(2.2809\) | |
| 82110.o4 | 82110q1 | \([1, 1, 0, -27832, -592064]\) | \(2384412229264108681/1234309176360960\) | \(1234309176360960\) | \([2]\) | \(507904\) | \(1.5878\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 82110.o have rank \(0\).
Complex multiplication
The elliptic curves in class 82110.o do not have complex multiplication.Modular form 82110.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
