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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 162450dy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 162450.bt4 | 162450dy1 | \([1, -1, 0, 758233683, -17087968861659]\) | \(89962967236397039/287450726400000\) | \(-154039666791630182400000000000\) | \([2]\) | \(165888000\) | \(4.2838\) | \(\Gamma_0(N)\)-optimal |
| 162450.bt3 | 162450dy2 | \([1, -1, 0, -7143334317, -200206807261659]\) | \(75224183150104868881/11219310000000000\) | \(6012226149768409218750000000000\) | \([2]\) | \(331776000\) | \(4.6304\) | |
| 162450.bt2 | 162450dy3 | \([1, -1, 0, -268161496317, -53449338913531659]\) | \(-3979640234041473454886161/1471455901872240\) | \(-788526714340484367147933750000\) | \([2]\) | \(829440000\) | \(5.0885\) | |
| 162450.bt1 | 162450dy4 | \([1, -1, 0, -4290584321817, -3420760270938474159]\) | \(16300610738133468173382620881/2228489100\) | \(1194207169736273217187500\) | \([2]\) | \(1658880000\) | \(5.4351\) |
Rank
sage: E.rank()
The elliptic curves in class 162450dy have rank \(1\).
Complex multiplication
The elliptic curves in class 162450dy do not have complex multiplication.Modular form 162450.2.a.dy
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
