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SageMath
sage: E = EllipticCurve("cp1")
sage: E.isogeny_class()
Elliptic curves in class 40425.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 40425.cp1 | 40425cg6 | [1, 0, 1, -3735270026, 87867764877323] | [2] | 8847360 | |
| 40425.cp2 | 40425cg4 | [1, 0, 1, -233454401, 1372918939823] | [2, 2] | 4423680 | |
| 40425.cp3 | 40425cg5 | [1, 0, 1, -232296776, 1387208662823] | [2] | 8847360 | |
| 40425.cp4 | 40425cg3 | [1, 0, 1, -31170151, -35325184177] | [2] | 4423680 | |
| 40425.cp5 | 40425cg2 | [1, 0, 1, -14663276, 21227369573] | [2, 2] | 2211840 | |
| 40425.cp6 | 40425cg1 | [1, 0, 1, 42849, 991741573] | [2] | 1105920 | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40425.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 40425.cp do not have complex multiplication.Modular form 40425.2.a.cp
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
