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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 242550gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 242550.gk7 | 242550gk1 | \([1, -1, 0, -12927042, 8136658116]\) | \(178272935636041/81841914000\) | \(109675992171806156250000\) | \([2]\) | \(21233664\) | \(3.1158\) | \(\Gamma_0(N)\)-optimal |
| 242550.gk5 | 242550gk2 | \([1, -1, 0, -173671542, 880497059616]\) | \(432288716775559561/270140062500\) | \(362013520114415039062500\) | \([2, 2]\) | \(42467328\) | \(3.4624\) | |
| 242550.gk4 | 242550gk3 | \([1, -1, 0, -526416417, -4648430576259]\) | \(12038605770121350841/757333463040\) | \(1014899272319338560000000\) | \([2]\) | \(63700992\) | \(3.6651\) | |
| 242550.gk2 | 242550gk4 | \([1, -1, 0, -2778327792, 56367489153366]\) | \(1769857772964702379561/691787250\) | \(927061077964957031250\) | \([2]\) | \(84934656\) | \(3.8090\) | |
| 242550.gk6 | 242550gk5 | \([1, -1, 0, -140927292, 1222641727866]\) | \(-230979395175477481/348191894531250\) | \(-466610439962287902832031250\) | \([2]\) | \(84934656\) | \(3.8090\) | |
| 242550.gk3 | 242550gk6 | \([1, -1, 0, -558168417, -4056033512259]\) | \(14351050585434661561/3001282273281600\) | \(4022005290709762073025000000\) | \([2, 2]\) | \(127401984\) | \(4.0117\) | |
| 242550.gk1 | 242550gk7 | \([1, -1, 0, -2827113417, 54285349272741]\) | \(1864737106103260904761/129177711985836360\) | \(173110488698131117780618125000\) | \([2]\) | \(254803968\) | \(4.3583\) | |
| 242550.gk8 | 242550gk8 | \([1, -1, 0, 1202744583, -24484385225259]\) | \(143584693754978072519/276341298967965000\) | \(-370323770071619708027578125000\) | \([2]\) | \(254803968\) | \(4.3583\) |
Rank
sage: E.rank()
The elliptic curves in class 242550gk have rank \(0\).
Complex multiplication
The elliptic curves in class 242550gk do not have complex multiplication.Modular form 242550.2.a.gk
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.
