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SageMath
sage: E = EllipticCurve("441.f1")
sage: E.isogeny_class()
Elliptic curves in class 441.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 441.f1 | 441c5 | [1, -1, 0, -345753, -78165914] | [2] | 1536 | |
| 441.f2 | 441c3 | [1, -1, 0, -21618, -1216265] | [2, 2] | 768 | |
| 441.f3 | 441c4 | [1, -1, 0, -17208, 867901] | [2] | 768 | |
| 441.f4 | 441c6 | [1, -1, 0, -15003, -1979636] | [2] | 1536 | |
| 441.f5 | 441c2 | [1, -1, 0, -1773, -5720] | [2, 2] | 384 | |
| 441.f6 | 441c1 | [1, -1, 0, 432, -869] | [2] | 192 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 441.f have rank \(1\).
Modular form 441.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
