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SageMath
E = EllipticCurve("qm1")
E.isogeny_class()
Elliptic curves in class 235200.qm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 235200.qm1 | 235200qm4 | \([0, 1, 0, -8508033, -8995355937]\) | \(282678688658/18600435\) | \(4481684638341120000000\) | \([2]\) | \(14155776\) | \(2.9041\) | |
| 235200.qm2 | 235200qm2 | \([0, 1, 0, -1648033, 642944063]\) | \(4108974916/893025\) | \(107585022182400000000\) | \([2, 2]\) | \(7077888\) | \(2.5576\) | |
| 235200.qm3 | 235200qm1 | \([0, 1, 0, -1550033, 742218063]\) | \(13674725584/945\) | \(28461646080000000\) | \([2]\) | \(3538944\) | \(2.2110\) | \(\Gamma_0(N)\)-optimal |
| 235200.qm4 | 235200qm3 | \([0, 1, 0, 3643967, 3929276063]\) | \(22208984782/40516875\) | \(-9762344605440000000000\) | \([2]\) | \(14155776\) | \(2.9041\) |
Rank
sage: E.rank()
The elliptic curves in class 235200.qm have rank \(1\).
Complex multiplication
The elliptic curves in class 235200.qm do not have complex multiplication.Modular form 235200.2.a.qm
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.
