## Defining polynomial

\(x^{15} + 2401 x^{3} - 67228\) |

## Invariants

Base field: | $\Q_{7}$ |

Degree $d$: | $15$ |

Ramification exponent $e$: | $3$ |

Residue field degree $f$: | $5$ |

Discriminant exponent $c$: | $10$ |

Discriminant root field: | $\Q_{7}$ |

Root number: | $1$ |

$\card{ \Gal(K/\Q_{ 7 }) }$: | $15$ |

This field is Galois and abelian over $\Q_{7}.$ | |

Visible slopes: | None |

## Intermediate fields

7.3.2.1, 7.5.0.1 |

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

## Unramified/totally ramified tower

Unramified subfield: | 7.5.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{5} + x + 4 \) |

Relative Eisenstein polynomial: | \( x^{3} + 7 t \) $\ \in\Q_{7}(t)[x]$ |

## Ramification polygon

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

Associated inertia: | $1$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $C_{15}$ (as 15T1) |

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

Wild inertia group: | $C_1$ |

Unramified degree: | $5$ |

Tame degree: | $3$ |

Wild slopes: | None |

Galois mean slope: | $2/3$ |

Galois splitting model: | Not computed |