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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 16650a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16650.u2 | 16650a1 | \([1, -1, 0, -942, 13716]\) | \(-219256227/59200\) | \(-24975000000\) | \([2]\) | \(13824\) | \(0.71136\) | \(\Gamma_0(N)\)-optimal |
16650.u1 | 16650a2 | \([1, -1, 0, -15942, 778716]\) | \(1062144635427/54760\) | \(23101875000\) | \([2]\) | \(27648\) | \(1.0579\) |
Rank
sage: E.rank()
The elliptic curves in class 16650a have rank \(1\).
Complex multiplication
The elliptic curves in class 16650a do not have complex multiplication.Modular form 16650.2.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 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.