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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 51842f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51842.k2 | 51842f1 | \([1, 1, 0, -5560594, 5056226148]\) | \(-3183010111/8464\) | \(-50562091580996137072\) | \([2]\) | \(1892352\) | \(2.6556\) | \(\Gamma_0(N)\)-optimal |
| 51842.k1 | 51842f2 | \([1, 1, 0, -89026214, 323277248960]\) | \(13062552753151/92\) | \(549587951967349316\) | \([2]\) | \(3784704\) | \(3.0021\) |
Rank
sage: E.rank()
The elliptic curves in class 51842f have rank \(1\).
Complex multiplication
The elliptic curves in class 51842f do not have complex multiplication.Modular form 51842.2.a.f
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.
