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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 41280cd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
41280.e1 | 41280cd1 | \([0, -1, 0, -1281, 14625]\) | \(3550014724/725625\) | \(47554560000\) | \([2]\) | \(36864\) | \(0.76428\) | \(\Gamma_0(N)\)-optimal |
41280.e2 | 41280cd2 | \([0, -1, 0, 2719, 84225]\) | \(16954370638/33698025\) | \(-4416867532800\) | \([2]\) | \(73728\) | \(1.1109\) |
Rank
sage: E.rank()
The elliptic curves in class 41280cd have rank \(1\).
Complex multiplication
The elliptic curves in class 41280cd do not have complex multiplication.Modular form 41280.2.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 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.