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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 850.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
850.e1 | 850b4 | \([1, 1, 0, -2825, -41125]\) | \(159661140625/48275138\) | \(754299031250\) | \([2]\) | \(1728\) | \(0.98421\) | |
850.e2 | 850b3 | \([1, 1, 0, -2575, -51375]\) | \(120920208625/19652\) | \(307062500\) | \([2]\) | \(864\) | \(0.63763\) | |
850.e3 | 850b2 | \([1, 1, 0, -1075, 13125]\) | \(8805624625/2312\) | \(36125000\) | \([2]\) | \(576\) | \(0.43490\) | |
850.e4 | 850b1 | \([1, 1, 0, -75, 125]\) | \(3048625/1088\) | \(17000000\) | \([2]\) | \(288\) | \(0.088327\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 850.e have rank \(0\).
Complex multiplication
The elliptic curves in class 850.e do not have complex multiplication.Modular form 850.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.