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SageMath
sage: E = EllipticCurve("do1")
sage: E.isogeny_class()
Elliptic curves in class 190575do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 190575.dx6 | 190575do1 | [1, -1, 0, 952308, -103907201909] | [2] | 22118400 | \(\Gamma_0(N)\)-optimal |
| 190575.dx5 | 190575do2 | [1, -1, 0, -325883817, -2224093144784] | [2, 2] | 44236800 | |
| 190575.dx4 | 190575do3 | [1, -1, 0, -692740692, 3701012243341] | [2] | 88473600 | |
| 190575.dx2 | 190575do4 | [1, -1, 0, -5188404942, -143845020910409] | [2, 2] | 88473600 | |
| 190575.dx3 | 190575do5 | [1, -1, 0, -5162677317, -145342188592034] | [2] | 176947200 | |
| 190575.dx1 | 190575do6 | [1, -1, 0, -83014470567, -9206145580482284] | [2] | 176947200 |
Rank
sage: E.rank()
The elliptic curves in class 190575do have rank \(0\).
Complex multiplication
The elliptic curves in class 190575do do not have complex multiplication.Modular form 190575.2.a.do
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
