## Defining polynomial

\(x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449\) |

## Invariants

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

Degree $d$: | $4$ |

Ramification exponent $e$: | $2$ |

Residue field degree $f$: | $2$ |

Discriminant exponent $c$: | $2$ |

Discriminant root field: | $\Q_{5}$ |

Root number: | $-1$ |

$\card{ \Gal(K/\Q_{ 5 }) }$: | $4$ |

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

Visible slopes: | None |

## Intermediate fields

$\Q_{5}(\sqrt{5})$, $\Q_{5}(\sqrt{5\cdot 2})$, $\Q_{5}(\sqrt{2})$ |

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

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{5}(\sqrt{2})$ $\cong \Q_{5}(t)$ where $t$ is a root of \( x^{2} + 4 x + 2 \) |

Relative Eisenstein polynomial: | \( x^{2} + 20 x + 5 \) $\ \in\Q_{5}(t)[x]$ |

## Ramification polygon

Residual polynomials: | $z + 2$ |

Associated inertia: | $1$ |

Indices of inseparability: | $[0]$ |

## Invariants of the Galois closure

Galois group: | $C_2^2$ (as 4T2) |

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

Wild inertia group: | $C_1$ |

Unramified degree: | $2$ |

Tame degree: | $2$ |

Wild slopes: | None |

Galois mean slope: | $1/2$ |

Galois splitting model: | $x^{4} + 15 x^{2} + 100$ |