Curve name | $X_{2}$ |

Index | $2$ |

Level | $2$ |

Genus | $0$ |

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

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

Images in lower levels | |

Meaning/Special name | Elliptic curves whose discriminant is a square |

Chosen covering | $X_{1}$ |

Curves that $X_{2}$ minimally covers | $X_{1}$ |

Curves that minimally cover $X_{2}$ | $X_{8}$, $X_{21}$, $X_{2a}$, $X_{2b}$ |

Curves that minimally cover $X_{2}$ and have infinitely many rational points. | $X_{8}$, $X_{2a}$, $X_{2b}$ |

Model | \[\mathbb{P}^{1}, \mathbb{Q}(X_{2}) = \mathbb{Q}(f_{2}), f_{1} = f_{2}^{2} + 1728\] |

Info about rational points | None |

Comments on finding rational points | None |

Elliptic curve whose $2$-adic image is the subgroup | $y^2 = x^3 - x^2 - 2x + 1$, with conductor $196$ |

Generic density of odd order reductions | $5/7$ |