## Defining polynomial

\(x^{9} + 1444 x^{3} - 116603\) |

## Invariants

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

Degree $d$: | $9$ |

Ramification exponent $e$: | $3$ |

Residue field degree $f$: | $3$ |

Discriminant exponent $c$: | $6$ |

Discriminant root field: | $\Q_{19}$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 19 }) }$: | $9$ |

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

Visible slopes: | None |

## Intermediate fields

19.3.0.1 |

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

## Unramified/totally ramified tower

Unramified subfield: | 19.3.0.1 $\cong \Q_{19}(t)$ where $t$ is a root of \( x^{3} + 4 x + 17 \) |

Relative Eisenstein polynomial: | \( x^{3} + 19 t \) $\ \in\Q_{19}(t)[x]$ |

## Ramification polygon

Residual polynomials: | $z^{2} + 3z + 3$ |

Associated inertia: | $1$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $C_9$ (as 9T1) |

Inertia group: | Intransitive group isomorphic to $C_3$ |

Wild inertia group: | $C_1$ |

Unramified degree: | $3$ |

Tame degree: | $3$ |

Wild slopes: | None |

Galois mean slope: | $2/3$ |

Galois splitting model: | Not computed |