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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 9702x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9702.x4 | 9702x1 | \([1, -1, 0, -891, 405]\) | \(912673/528\) | \(45284511888\) | \([2]\) | \(9216\) | \(0.73416\) | \(\Gamma_0(N)\)-optimal |
9702.x2 | 9702x2 | \([1, -1, 0, -9711, -364743]\) | \(1180932193/4356\) | \(373597223076\) | \([2, 2]\) | \(18432\) | \(1.0807\) | |
9702.x1 | 9702x3 | \([1, -1, 0, -155241, -23504013]\) | \(4824238966273/66\) | \(5660563986\) | \([2]\) | \(36864\) | \(1.4273\) | |
9702.x3 | 9702x4 | \([1, -1, 0, -5301, -700785]\) | \(-192100033/2371842\) | \(-203423687964882\) | \([2]\) | \(36864\) | \(1.4273\) |
Rank
sage: E.rank()
The elliptic curves in class 9702x have rank \(0\).
Complex multiplication
The elliptic curves in class 9702x do not have complex multiplication.Modular form 9702.2.a.x
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.