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SageMath
E = EllipticCurve("gn1")
E.isogeny_class()
Elliptic curves in class 388080gn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
388080.gn2 | 388080gn1 | \([0, 0, 0, 1932, 13867]\) | \(199344128/136125\) | \(-544602366000\) | \([2]\) | \(368640\) | \(0.94058\) | \(\Gamma_0(N)\)-optimal |
388080.gn1 | 388080gn2 | \([0, 0, 0, -8463, 115738]\) | \(1047213232/515625\) | \(33006204000000\) | \([2]\) | \(737280\) | \(1.2872\) |
Rank
sage: E.rank()
The elliptic curves in class 388080gn have rank \(1\).
Complex multiplication
The elliptic curves in class 388080gn do not have complex multiplication.Modular form 388080.2.a.gn
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.