| Curve name | $X_{8}$ |
| Index | $6$ |
| Level | $2$ |
| Genus | $0$ |
| Does the subgroup contain $-I$? | Yes |
| Generating matrices | |
| Images in lower levels | |
| Meaning/Special name | Elliptic curves with full $2$-torsion over $\mathbb{Q}$ |
| Chosen covering | $X_{6}$ |
| Curves that $X_{8}$ minimally covers | $X_{2}$, $X_{6}$ |
| Curves that minimally cover $X_{8}$ | $X_{24}$, $X_{25}$, $X_{38}$, $X_{46}$, $X_{8a}$, $X_{8b}$, $X_{8c}$, $X_{8d}$ |
| Curves that minimally cover $X_{8}$ and have infinitely many rational points. | $X_{24}$, $X_{25}$, $X_{38}$, $X_{46}$, $X_{8a}$, $X_{8b}$, $X_{8c}$, $X_{8d}$ |
| Model | \[\mathbb{P}^{1}, \mathbb{Q}(X_{8}) = \mathbb{Q}(f_{8}), f_{6} = \frac{8f_{8}^{2} + 24}{f_{8} - 1}\] |
| Info about rational points | None |
| Comments on finding rational points | None |
| Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy + y = x^3 - x^2 - 68x + 182$, with conductor $315$ |
| Generic density of odd order reductions | $1/7$ |