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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 265200ds
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 265200.ds4 | 265200ds1 | \([0, -1, 0, 631992, -33001488]\) | \(436192097814719/259683840000\) | \(-16619765760000000000\) | \([2]\) | \(6635520\) | \(2.3774\) | \(\Gamma_0(N)\)-optimal |
| 265200.ds3 | 265200ds2 | \([0, -1, 0, -2568008, -263401488]\) | \(29263955267177281/16463793153600\) | \(1053682761830400000000\) | \([2, 2]\) | \(13271040\) | \(2.7240\) | |
| 265200.ds2 | 265200ds3 | \([0, -1, 0, -25688008, 49860758512]\) | \(29291056630578924481/175463302795560\) | \(11229651378915840000000\) | \([2]\) | \(26542080\) | \(3.0706\) | |
| 265200.ds1 | 265200ds4 | \([0, -1, 0, -30648008, -65184361488]\) | \(49745123032831462081/97939634471640\) | \(6268136606184960000000\) | \([2]\) | \(26542080\) | \(3.0706\) |
Rank
sage: E.rank()
The elliptic curves in class 265200ds have rank \(1\).
Complex multiplication
The elliptic curves in class 265200ds do not have complex multiplication.Modular form 265200.2.a.ds
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
