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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 189630n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
189630.eo2 | 189630n1 | \([1, -1, 1, -12137, -808279]\) | \(-2305199161/1981440\) | \(-169940422794240\) | \([2]\) | \(737280\) | \(1.4280\) | \(\Gamma_0(N)\)-optimal |
189630.eo1 | 189630n2 | \([1, -1, 1, -223817, -40688791]\) | \(14457238157881/4437600\) | \(380595738549600\) | \([2]\) | \(1474560\) | \(1.7746\) |
Rank
sage: E.rank()
The elliptic curves in class 189630n have rank \(1\).
Complex multiplication
The elliptic curves in class 189630n do not have complex multiplication.Modular form 189630.2.a.n
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.