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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 462462.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
462462.r1 | 462462r3 | \([1, 1, 0, -2472516, 1495371354]\) | \(8020417344913/187278\) | \(39032926498307742\) | \([2]\) | \(11796480\) | \(2.2959\) | \(\Gamma_0(N)\)-optimal* |
462462.r2 | 462462r2 | \([1, 1, 0, -160206, 21504960]\) | \(2181825073/298116\) | \(62134046262612324\) | \([2, 2]\) | \(5898240\) | \(1.9493\) | \(\Gamma_0(N)\)-optimal* |
462462.r3 | 462462r1 | \([1, 1, 0, -41626, -2946236]\) | \(38272753/4368\) | \(910388956228752\) | \([2]\) | \(2949120\) | \(1.6027\) | \(\Gamma_0(N)\)-optimal* |
462462.r4 | 462462r4 | \([1, 1, 0, 254824, 114886710]\) | \(8780064047/32388174\) | \(-6750420311816667486\) | \([2]\) | \(11796480\) | \(2.2959\) |
Rank
sage: E.rank()
The elliptic curves in class 462462.r have rank \(1\).
Complex multiplication
The elliptic curves in class 462462.r do not have complex multiplication.Modular form 462462.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.