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SageMath
E = EllipticCurve("gj1")
E.isogeny_class()
Elliptic curves in class 481650gj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
481650.gj2 | 481650gj1 | \([1, 1, 1, -6151688, -5902240219]\) | \(-341370886042369/1817528220\) | \(-137075962032030937500\) | \([2]\) | \(25804800\) | \(2.7087\) | \(\Gamma_0(N)\)-optimal* |
481650.gj1 | 481650gj2 | \([1, 1, 1, -98552438, -376614049219]\) | \(1403607530712116449/39475350\) | \(2977187104033593750\) | \([2]\) | \(51609600\) | \(3.0553\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 481650gj have rank \(0\).
Complex multiplication
The elliptic curves in class 481650gj do not have complex multiplication.Modular form 481650.2.a.gj
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.