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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 258570.eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 258570.eg1 | 258570eg4 | \([1, -1, 1, -116539052, -483380508009]\) | \(49745123032831462081/97939634471640\) | \(344624477751703781438040\) | \([2]\) | \(61931520\) | \(3.4045\) | |
| 258570.eg2 | 258570eg3 | \([1, -1, 1, -97678652, 369672220311]\) | \(29291056630578924481/175463302795560\) | \(617410401996330608501160\) | \([2]\) | \(61931520\) | \(3.4045\) | |
| 258570.eg3 | 258570eg2 | \([1, -1, 1, -9764852, -1956995049]\) | \(29263955267177281/16463793153600\) | \(57931869441624514689600\) | \([2, 2]\) | \(30965760\) | \(3.0579\) | |
| 258570.eg4 | 258570eg1 | \([1, -1, 1, 2403148, -243740649]\) | \(436192097814719/259683840000\) | \(-913760891832522240000\) | \([4]\) | \(15482880\) | \(2.7113\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 258570.eg have rank \(1\).
Complex multiplication
The elliptic curves in class 258570.eg do not have complex multiplication.Modular form 258570.2.a.eg
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
