## Defining polynomial

\( x^{11} + 44 x^{4} + 11 \) |

## Invariants

Base field: | $\Q_{11}$ |

Degree $d$ : | $11$ |

Ramification exponent $e$ : | $11$ |

Residue field degree $f$ : | $1$ |

Discriminant exponent $c$ : | $14$ |

Discriminant root field: | $\Q_{11}$ |

Root number: | $1$ |

$|\Aut(K/\Q_{ 11 })|$: | $1$ |

This field is not Galois over $\Q_{11}$. |

## Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$. |

## Unramified/totally ramified tower

Unramified subfield: | $\Q_{11}$ |

Relative Eisenstein polynomial: | \( x^{11} + 44 x^{4} + 11 \) |

## Invariants of the Galois closure

Galois group: | $C_{11}:C_5$ (as 11T3) |

Inertia group: | $C_{11}:C_5$ |

Unramified degree: | $1$ |

Tame degree: | $5$ |

Wild slopes: | [7/5] |

Galois mean slope: | $74/55$ |

Galois splitting model: | Not computed |