21、椭圆曲线相关问题探讨

椭圆曲线相关问题探讨

1. 椭圆曲线在有理数域上的练习

1.1 椭圆曲线的挠群验证

有一系列椭圆曲线需要验证其挠群，具体如下：
| 椭圆曲线方程 | 对应挠群 |
| — | — |
| (y^2 = x^3 - 2) | (0) |
| (y^2 = x^3 + 8) | (\mathbb{Z} 2) |
| (y^2 = x^3 + 4) | (\mathbb{Z}_3) |
| (y^2 = x^3 + 4x) | (\mathbb{Z}_4) |
| (y^2 = x^3 - 432x + 8208) | (\mathbb{Z}_5) |
| (y^2 = x^3 + 1) | (\mathbb{Z}_6) |
| (y^2 = x^3 - 1323x + 6395814) | (\mathbb{Z}_7) |
| (y^2 = x^3 - 44091x + 3304854) | (\mathbb{Z}_8) |
| (y^2 = x^3 - 219x + 1654) | (\mathbb{Z}_9) |
| (y^2 = x^3 - 58347x + 3954150) | (\mathbb{Z} {10}) |
| (y^2 = x^3 - 33339627x + 73697852646) | (\mathbb{Z}_{12}) |
| (y^2 = x^3 - x) | (\mathbb{Z}_2 \oplus \mathbb{Z}_2) |
| (y^2 = x^3 - 12987x - 263466) | (\