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SageMath
sage: E = EllipticCurve("a1")
sage: E.isogeny_class()
Elliptic curves in class 126.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 126.a1 | 126b3 | [1, -1, 0, -12096, -509036] | [2] | 128 | |
| 126.a2 | 126b5 | [1, -1, 0, -8226, 286474] | [2] | 256 | |
| 126.a3 | 126b4 | [1, -1, 0, -936, -3668] | [2, 2] | 128 | |
| 126.a4 | 126b2 | [1, -1, 0, -756, -7808] | [2, 2] | 64 | |
| 126.a5 | 126b1 | [1, -1, 0, -36, -176] | [2] | 32 | \(\Gamma_0(N)\)-optimal |
| 126.a6 | 126b6 | [1, -1, 0, 3474, -31010] | [2] | 256 |
Rank
sage: E.rank()
The elliptic curves in class 126.a have rank \(0\).
Complex multiplication
The elliptic curves in class 126.a do not have complex multiplication.Modular form 126.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
