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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 388080dy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.dy3 | 388080dy1 | \([0, 0, 0, -6992643, 4905464578]\) | \(107639597521009/32699842560\) | \(11487390325481102376960\) | \([2]\) | \(23592960\) | \(2.9380\) | \(\Gamma_0(N)\)-optimal |
| 388080.dy2 | 388080dy2 | \([0, 0, 0, -43119363, -105201552638]\) | \(25238585142450289/995844326400\) | \(349838151660281423462400\) | \([2, 2]\) | \(47185920\) | \(3.2845\) | |
| 388080.dy4 | 388080dy3 | \([0, 0, 0, 18973437, -383290366718]\) | \(2150235484224911/181905111732960\) | \(-63902907692677394629263360\) | \([2]\) | \(94371840\) | \(3.6311\) | |
| 388080.dy1 | 388080dy4 | \([0, 0, 0, -683239683, -6873961840382]\) | \(100407751863770656369/166028940000\) | \(58325639813290967040000\) | \([2]\) | \(94371840\) | \(3.6311\) |
Rank
sage: E.rank()
The elliptic curves in class 388080dy have rank \(0\).
Complex multiplication
The elliptic curves in class 388080dy do not have complex multiplication.Modular form 388080.2.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.
