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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 25410cu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.cs7 | 25410cu1 | \([1, 0, 0, -141875, -9363375]\) | \(178272935636041/81841914000\) | \(144987943007754000\) | \([4]\) | \(276480\) | \(1.9878\) | \(\Gamma_0(N)\)-optimal |
| 25410.cs5 | 25410cu2 | \([1, 0, 0, -1906055, -1012476123]\) | \(432288716775559561/270140062500\) | \(478569599262562500\) | \([2, 2]\) | \(552960\) | \(2.3344\) | |
| 25410.cs4 | 25410cu3 | \([1, 0, 0, -5777450, 5344285860]\) | \(12038605770121350841/757333463040\) | \(1341662427116605440\) | \([4]\) | \(829440\) | \(2.5371\) | |
| 25410.cs6 | 25410cu4 | \([1, 0, 0, -1546685, -1405842525]\) | \(-230979395175477481/348191894531250\) | \(-616843180867675781250\) | \([2]\) | \(1105920\) | \(2.6809\) | |
| 25410.cs2 | 25410cu5 | \([1, 0, 0, -30492305, -64811268873]\) | \(1769857772964702379561/691787250\) | \(1225543312397250\) | \([2]\) | \(1105920\) | \(2.6809\) | |
| 25410.cs3 | 25410cu6 | \([1, 0, 0, -6125930, 4663146852]\) | \(14351050585434661561/3001282273281600\) | \(5316954625337024577600\) | \([2, 2]\) | \(1658880\) | \(2.8837\) | |
| 25410.cs8 | 25410cu7 | \([1, 0, 0, 13200190, 28152113100]\) | \(143584693754978072519/276341298967965000\) | \(-489555467940987043365000\) | \([2]\) | \(3317760\) | \(3.2302\) | |
| 25410.cs1 | 25410cu8 | \([1, 0, 0, -31027730, -62417321988]\) | \(1864737106103260904761/129177711985836360\) | \(228846196623340247757960\) | \([2]\) | \(3317760\) | \(3.2302\) |
Rank
sage: E.rank()
The elliptic curves in class 25410cu have rank \(1\).
Complex multiplication
The elliptic curves in class 25410cu do not have complex multiplication.Modular form 25410.2.a.cu
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.
