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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 8619.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8619.e1 | 8619d2 | \([1, 1, 1, -55351, -1700704]\) | \(3885442650361/1996623837\) | \(9637321906046133\) | \([2]\) | \(96768\) | \(1.7592\) | |
8619.e2 | 8619d1 | \([1, 1, 1, -44366, -3612094]\) | \(2000852317801/2094417\) | \(10109350825353\) | \([2]\) | \(48384\) | \(1.4126\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 8619.e have rank \(0\).
Complex multiplication
The elliptic curves in class 8619.e do not have complex multiplication.Modular form 8619.2.a.e
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 LMFDB 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 LMFDB labels.