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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 29760cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
29760.cb2 | 29760cf1 | \([0, 1, 0, 159, 11295]\) | \(1685159/209250\) | \(-54853632000\) | \([]\) | \(23040\) | \(0.74002\) | \(\Gamma_0(N)\)-optimal |
29760.cb1 | 29760cf2 | \([0, 1, 0, -33441, 2343135]\) | \(-15777367606441/3574920\) | \(-937143828480\) | \([]\) | \(69120\) | \(1.2893\) |
Rank
sage: E.rank()
The elliptic curves in class 29760cf have rank \(1\).
Complex multiplication
The elliptic curves in class 29760cf do not have complex multiplication.Modular form 29760.2.a.cf
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.