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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 4998r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4998.t2 | 4998r1 | \([1, 0, 1, -180, 23986]\) | \(-1865409391/724451328\) | \(-248486805504\) | \([2]\) | \(8064\) | \(0.86570\) | \(\Gamma_0(N)\)-optimal |
4998.t1 | 4998r2 | \([1, 0, 1, -13620, 604594]\) | \(814544990575471/9268826496\) | \(3179207488128\) | \([2]\) | \(16128\) | \(1.2123\) |
Rank
sage: E.rank()
The elliptic curves in class 4998r have rank \(1\).
Complex multiplication
The elliptic curves in class 4998r do not have complex multiplication.Modular form 4998.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.