## Defining polynomial

\( x^{6} - 10 x^{4} + 25 x^{2} - 500 \) |

## Invariants

Base field: | $\Q_{5}$ |

Degree $d$: | $6$ |

Ramification exponent $e$: | $2$ |

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

Discriminant exponent $c$: | $3$ |

Discriminant root field: | $\Q_{5}(\sqrt{5})$ |

Root number: | $1$ |

$|\Gal(K/\Q_{ 5 })|$: | $6$ |

This field is Galois and abelian over $\Q_{5}.$ |

## Intermediate fields

$\Q_{5}(\sqrt{5})$, 5.3.0.1 |

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

## Unramified/totally ramified tower

Unramified subfield: | 5.3.0.1 $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{3} - x + 2 \) |

Relative Eisenstein polynomial: | $ x^{2} - 5 t^{2} \in\Q_{5}(t)[x]$ |

## Invariants of the Galois closure

Galois group: | $C_6$ (as 6T1) |

Inertia group: | Intransitive group isomorphic to $C_2$ |

Unramified degree: | $3$ |

Tame degree: | $2$ |

Wild slopes: | None |

Galois mean slope: | $1/2$ |

Galois splitting model: | $x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$ |