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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 9702.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9702.r1 | 9702p4 | \([1, -1, 0, -17748936, 28785485182]\) | \(7209828390823479793/49509306\) | \(4246221129022026\) | \([2]\) | \(294912\) | \(2.5985\) | |
| 9702.r2 | 9702p3 | \([1, -1, 0, -1546596, 63354514]\) | \(4770223741048753/2740574865798\) | \(235048475549590029558\) | \([2]\) | \(294912\) | \(2.5985\) | |
| 9702.r3 | 9702p2 | \([1, -1, 0, -1110006, 449387392]\) | \(1763535241378513/4612311396\) | \(395580057279014916\) | \([2, 2]\) | \(147456\) | \(2.2519\) | |
| 9702.r4 | 9702p1 | \([1, -1, 0, -42786, 12467524]\) | \(-100999381393/723148272\) | \(-62021622197292912\) | \([2]\) | \(73728\) | \(1.9053\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9702.r have rank \(1\).
Complex multiplication
The elliptic curves in class 9702.r do not have complex multiplication.Modular form 9702.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
