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 | $1/7$ |