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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 221760.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.a1 | 221760nr2 | \([0, 0, 0, -43308, -3417552]\) | \(1740992427/29645\) | \(152961688535040\) | \([2]\) | \(983040\) | \(1.5194\) | |
221760.a2 | 221760nr1 | \([0, 0, 0, -108, -151632]\) | \(-27/1925\) | \(-9932577177600\) | \([2]\) | \(491520\) | \(1.1728\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.a have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.a do not have complex multiplication.Modular form 221760.2.a.a
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.