Curve name | $X_{6}$ |

Index | $3$ |

Level | $2$ |

Genus | $0$ |

Does the subgroup contain $-I$? | Yes |

Generating matrices | $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right]$ |

Images in lower levels | |

Meaning/Special name | Elliptic curves with a rational $2$-torsion point |

Chosen covering | $X_{1}$ |

Curves that $X_{6}$ minimally covers | $X_{1}$ |

Curves that minimally cover $X_{6}$ | $X_{8}$, $X_{9}$, $X_{10}$, $X_{11}$, $X_{12}$, $X_{13}$, $X_{14}$, $X_{15}$, $X_{16}$, $X_{17}$, $X_{18}$, $X_{19}$ |

Curves that minimally cover $X_{6}$ and have infinitely many rational points. | $X_{8}$, $X_{9}$, $X_{10}$, $X_{11}$, $X_{12}$, $X_{13}$, $X_{14}$, $X_{15}$, $X_{16}$, $X_{17}$, $X_{18}$, $X_{19}$ |

Model | \[\mathbb{P}^{1}, \mathbb{Q}(X_{6}) = \mathbb{Q}(f_{6}), f_{1} = \frac{f_{6}^{3}}{f_{6} + 16}\] |

Info about rational points | None |

Comments on finding rational points | None |

Elliptic curve whose $2$-adic image is the subgroup | $y^2 + xy + y = x^3 - x - 1$, with conductor $69$ |

Generic density of odd order reductions | $5/21$ |