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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 39039.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
39039.j1 | 39039y1 | \([1, 0, 0, -3016803, -2017053000]\) | \(1382084250541230782125/19771083137421\) | \(43437069652913937\) | \([2]\) | \(756288\) | \(2.3319\) | \(\Gamma_0(N)\)-optimal |
39039.j2 | 39039y2 | \([1, 0, 0, -2930288, -2138156697]\) | \(-1266556547153680328125/165777947457789051\) | \(-364214150564762545047\) | \([2]\) | \(1512576\) | \(2.6784\) |
Rank
sage: E.rank()
The elliptic curves in class 39039.j have rank \(0\).
Complex multiplication
The elliptic curves in class 39039.j do not have complex multiplication.Modular form 39039.2.a.j
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.