## Defining polynomial

\(x^{4} + 2 x^{2} + 11 x + 2\) |

## Invariants

Base field: | $\Q_{19}$ |

Degree $d$: | $4$ |

Ramification exponent $e$: | $1$ |

Residue field degree $f$: | $4$ |

Discriminant exponent $c$: | $0$ |

Discriminant root field: | $\Q_{19}(\sqrt{2})$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 19 }) }$: | $4$ |

This field is Galois and abelian over $\Q_{19}.$ | |

Visible slopes: | None |

## Intermediate fields

$\Q_{19}(\sqrt{2})$ |

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Unramified/totally ramified tower

Unramified subfield: | 19.4.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{4} + 2 x^{2} + 11 x + 2 \) |

Relative Eisenstein polynomial: | \( x - 19 \) $\ \in\Q_{19}(t)[x]$ |

## Ramification polygon

The ramification polygon is trivial for unramified extensions.