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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 388416j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388416.j3 | 388416j1 | \([0, -1, 0, 1997183, -29318308319]\) | \(139233463487/58763045376\) | \(-371824279529280226983936\) | \([]\) | \(59719680\) | \(3.2018\) | \(\Gamma_0(N)\)-optimal |
| 388416.j2 | 388416j2 | \([0, -1, 0, -17978497, 792533103649]\) | \(-101566487155393/42823570577256\) | \(-270966951676447694694383616\) | \([]\) | \(179159040\) | \(3.7511\) | |
| 388416.j1 | 388416j3 | \([0, -1, 0, -7059960577, 228334161950881]\) | \(-6150311179917589675873/244053849830826\) | \(-1544255344478047673252315136\) | \([]\) | \(537477120\) | \(4.3004\) |
Rank
sage: E.rank()
The elliptic curves in class 388416j have rank \(1\).
Complex multiplication
The elliptic curves in class 388416j do not have complex multiplication.Modular form 388416.2.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 Cremona numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
