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SageMath
sage: E = EllipticCurve("e1")
sage: E.isogeny_class()
Elliptic curves in class 6864.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 6864.e1 | 6864r3 | [0, -1, 0, -109824, -13972032] | [2] | 16384 | |
| 6864.e2 | 6864r5 | [0, -1, 0, -48144, 3953280] | [4] | 32768 | |
| 6864.e3 | 6864r4 | [0, -1, 0, -7584, -167616] | [2, 4] | 16384 | |
| 6864.e4 | 6864r2 | [0, -1, 0, -6864, -216576] | [2, 2] | 8192 | |
| 6864.e5 | 6864r1 | [0, -1, 0, -384, -4032] | [2] | 4096 | \(\Gamma_0(N)\)-optimal |
| 6864.e6 | 6864r6 | [0, -1, 0, 21456, -1166592] | [4] | 32768 |
Rank
sage: E.rank()
The elliptic curves in class 6864.e have rank \(0\).
Complex multiplication
The elliptic curves in class 6864.e do not have complex multiplication.Modular form 6864.2.a.e
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.
