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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 9680r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
9680.g1 | 9680r1 | \([0, -1, 0, -1976, 93040]\) | \(-117649/440\) | \(-3192778096640\) | \([]\) | \(11520\) | \(1.0850\) | \(\Gamma_0(N)\)-optimal |
9680.g2 | 9680r2 | \([0, -1, 0, 17384, -2199184]\) | \(80062991/332750\) | \(-2414538435584000\) | \([]\) | \(34560\) | \(1.6343\) |
Rank
sage: E.rank()
The elliptic curves in class 9680r have rank \(1\).
Complex multiplication
The elliptic curves in class 9680r do not have complex multiplication.Modular form 9680.2.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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.