## Defining polynomial

\( x^{6} - x + 1 \) |

## Invariants

Base field: | $\Q_{2}$ |

Degree $d$ : | $6$ |

Ramification exponent $e$ : | $1$ |

Residue field degree $f$ : | $6$ |

Discriminant exponent $c$ : | $0$ |

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

Root number: | $1$ |

$|\Gal(K/\Q_{ 2 })|$: | $6$ |

This field is Galois and abelian over $\Q_{2}$. |

## Intermediate fields

$\Q_{2}(\sqrt{*})$, 2.3.0.1 |

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

## Unramified/totally ramified tower

Unramified subfield: | 2.6.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{6} - x + 1 \) |

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

## Invariants of the Galois closure

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

Inertia group: | Trivial |

Unramified degree: | $6$ |

Tame degree: | $1$ |

Wild slopes: | None |

Galois mean slope: | $0$ |

Galois splitting model: | \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 \) |