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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 48510eg
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48510.dz7 | 48510eg1 | \([1, -1, 1, -517082, 65196681]\) | \(178272935636041/81841914000\) | \(7019263498995594000\) | \([2]\) | \(884736\) | \(2.3111\) | \(\Gamma_0(N)\)-optimal |
| 48510.dz5 | 48510eg2 | \([1, -1, 1, -6946862, 7045365849]\) | \(432288716775559561/270140062500\) | \(23168865287322562500\) | \([2, 2]\) | \(1769472\) | \(2.6577\) | |
| 48510.dz4 | 48510eg3 | \([1, -1, 1, -21056657, -37183233279]\) | \(12038605770121350841/757333463040\) | \(64953553428437667840\) | \([2]\) | \(2654208\) | \(2.8604\) | |
| 48510.dz6 | 48510eg4 | \([1, -1, 1, -5637092, 9782261241]\) | \(-230979395175477481/348191894531250\) | \(-29863068157586425781250\) | \([2]\) | \(3538944\) | \(3.0042\) | |
| 48510.dz2 | 48510eg5 | \([1, -1, 1, -111133112, 450962139849]\) | \(1769857772964702379561/691787250\) | \(59331908989757250\) | \([2]\) | \(3538944\) | \(3.0042\) | |
| 48510.dz3 | 48510eg6 | \([1, -1, 1, -22326737, -32443802751]\) | \(14351050585434661561/3001282273281600\) | \(257408338605424772673600\) | \([2, 2]\) | \(5308416\) | \(3.2070\) | |
| 48510.dz8 | 48510eg7 | \([1, -1, 1, 48109783, -195884703759]\) | \(143584693754978072519/276341298967965000\) | \(-23700721284583661313765000\) | \([2]\) | \(10616832\) | \(3.5535\) | |
| 48510.dz1 | 48510eg8 | \([1, -1, 1, -113084537, 434305411089]\) | \(1864737106103260904761/129177711985836360\) | \(11079071276680391537959560\) | \([2]\) | \(10616832\) | \(3.5535\) |
Rank
sage: E.rank()
The elliptic curves in class 48510eg have rank \(0\).
Complex multiplication
The elliptic curves in class 48510eg do not have complex multiplication.Modular form 48510.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
