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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 286650dy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.dy3 | 286650dy1 | \([1, -1, 0, -149067, 21524341]\) | \(273359449/9360\) | \(12543295196250000\) | \([2]\) | \(2359296\) | \(1.8609\) | \(\Gamma_0(N)\)-optimal |
| 286650.dy2 | 286650dy2 | \([1, -1, 0, -369567, -56753159]\) | \(4165509529/1368900\) | \(1834456922451562500\) | \([2, 2]\) | \(4718592\) | \(2.2074\) | |
| 286650.dy4 | 286650dy3 | \([1, -1, 0, 1063683, -390700409]\) | \(99317171591/106616250\) | \(-142875971844785156250\) | \([2]\) | \(9437184\) | \(2.5540\) | |
| 286650.dy1 | 286650dy4 | \([1, -1, 0, -5330817, -4735211909]\) | \(12501706118329/2570490\) | \(3444702443270156250\) | \([2]\) | \(9437184\) | \(2.5540\) |
Rank
sage: E.rank()
The elliptic curves in class 286650dy have rank \(1\).
Complex multiplication
The elliptic curves in class 286650dy do not have complex multiplication.Modular form 286650.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
