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SageMath
E = EllipticCurve("ic1")
E.isogeny_class()
Elliptic curves in class 221760ic
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.nx2 | 221760ic1 | \([0, 0, 0, -321852, -82303846]\) | \(-79028701534867456/16987307596875\) | \(-792559823239800000\) | \([]\) | \(3840000\) | \(2.1556\) | \(\Gamma_0(N)\)-optimal |
221760.nx1 | 221760ic2 | \([0, 0, 0, -964452, 6892234634]\) | \(-2126464142970105856/438611057788643355\) | \(-20463837512186944370880\) | \([]\) | \(19200000\) | \(2.9604\) |
Rank
sage: E.rank()
The elliptic curves in class 221760ic have rank \(1\).
Complex multiplication
The elliptic curves in class 221760ic do not have complex multiplication.Modular form 221760.2.a.ic
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.