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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 9702.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9702.a1 | 9702t3 | \([1, -1, 0, -4438674, -3598272896]\) | \(112763292123580561/1932612\) | \(165752634638052\) | \([2]\) | \(288000\) | \(2.2701\) | |
| 9702.a2 | 9702t4 | \([1, -1, 0, -4434264, -3605783126]\) | \(-112427521449300721/466873642818\) | \(-40041941341639368978\) | \([2]\) | \(576000\) | \(2.6167\) | |
| 9702.a3 | 9702t1 | \([1, -1, 0, -19854, 789844]\) | \(10091699281/2737152\) | \(234754909627392\) | \([2]\) | \(57600\) | \(1.4654\) | \(\Gamma_0(N)\)-optimal |
| 9702.a4 | 9702t2 | \([1, -1, 0, 50706, 5094004]\) | \(168105213359/228637728\) | \(-19609371044813088\) | \([2]\) | \(115200\) | \(1.8120\) |
Rank
sage: E.rank()
The elliptic curves in class 9702.a have rank \(1\).
Complex multiplication
The elliptic curves in class 9702.a do not have complex multiplication.Modular form 9702.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
