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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 45630.cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45630.cd1 | 45630by1 | \([1, -1, 1, -26903, 1705487]\) | \(-16522921323/4000\) | \(-521295372000\) | \([]\) | \(123120\) | \(1.2371\) | \(\Gamma_0(N)\)-optimal |
45630.cd2 | 45630by2 | \([1, -1, 1, 11122, 5982877]\) | \(1601613/163840\) | \(-15565796400660480\) | \([]\) | \(369360\) | \(1.7864\) |
Rank
sage: E.rank()
The elliptic curves in class 45630.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 45630.cd do not have complex multiplication.Modular form 45630.2.a.cd
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.