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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 72450bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.l2 | 72450bg1 | \([1, -1, 0, 3933, 23841]\) | \(590589719/365148\) | \(-4159263937500\) | \([2]\) | \(122880\) | \(1.1105\) | \(\Gamma_0(N)\)-optimal |
72450.l1 | 72450bg2 | \([1, -1, 0, -16317, 206091]\) | \(42180533641/22862322\) | \(260416136531250\) | \([2]\) | \(245760\) | \(1.4571\) |
Rank
sage: E.rank()
The elliptic curves in class 72450bg have rank \(1\).
Complex multiplication
The elliptic curves in class 72450bg do not have complex multiplication.Modular form 72450.2.a.bg
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.