Show commands:
SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 106560.dx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106560.dx1 | 106560cz4 | \([0, 0, 0, -6040812, 2550912176]\) | \(127568139540190201/59114336463360\) | \(11296925622413410959360\) | \([2]\) | \(9289728\) | \(2.9262\) | |
| 106560.dx2 | 106560cz2 | \([0, 0, 0, -3060012, -2060203984]\) | \(16581570075765001/998001000\) | \(190720961150976000\) | \([2]\) | \(3096576\) | \(2.3769\) | |
| 106560.dx3 | 106560cz1 | \([0, 0, 0, -180012, -36139984]\) | \(-3375675045001/999000000\) | \(-190911873024000000\) | \([2]\) | \(1548288\) | \(2.0303\) | \(\Gamma_0(N)\)-optimal |
| 106560.dx4 | 106560cz3 | \([0, 0, 0, 1331988, 300733616]\) | \(1367594037332999/995878502400\) | \(-190315345543063142400\) | \([2]\) | \(4644864\) | \(2.5796\) |
Rank
sage: E.rank()
The elliptic curves in class 106560.dx have rank \(1\).
Complex multiplication
The elliptic curves in class 106560.dx do not have complex multiplication.Modular form 106560.2.a.dx
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
