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SageMath
E = EllipticCurve("mq1")
E.isogeny_class()
Elliptic curves in class 187200mq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.jd4 | 187200mq1 | \([0, 0, 0, -20775, -164500]\) | \(1360251712/771147\) | \(562166163000000\) | \([2]\) | \(786432\) | \(1.5202\) | \(\Gamma_0(N)\)-optimal |
| 187200.jd2 | 187200mq2 | \([0, 0, 0, -210900, 37100000]\) | \(22235451328/123201\) | \(5748065856000000\) | \([2, 2]\) | \(1572864\) | \(1.8667\) | |
| 187200.jd1 | 187200mq3 | \([0, 0, 0, -3369900, 2381078000]\) | \(11339065490696/351\) | \(131010048000000\) | \([2]\) | \(3145728\) | \(2.2133\) | |
| 187200.jd3 | 187200mq4 | \([0, 0, 0, -93900, 78050000]\) | \(-245314376/6908733\) | \(-2578670774784000000\) | \([2]\) | \(3145728\) | \(2.2133\) |
Rank
sage: E.rank()
The elliptic curves in class 187200mq have rank \(1\).
Complex multiplication
The elliptic curves in class 187200mq do not have complex multiplication.Modular form 187200.2.a.mq
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
