# Properties

 Label 11.12.8.2 Base $$\Q_{11}$$ Degree $$12$$ e $$3$$ f $$4$$ c $$8$$ Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

# Related objects

## Defining polynomial

 $$x^{12} - 176 x^{9} + 8228 x^{6} + 90508 x^{3} + 58564$$ x^12 - 176*x^9 + 8228*x^6 + 90508*x^3 + 58564

## Invariants

 Base field: $\Q_{11}$ Degree $d$: $12$ Ramification exponent $e$: $3$ Residue field degree $f$: $4$ Discriminant exponent $c$: $8$ Discriminant root field: $\Q_{11}(\sqrt{2})$ Root number: $1$ $\card{ \Aut(K/\Q_{ 11 }) }$: $6$ This field is not Galois over $\Q_{11}.$ Visible slopes: None

## Intermediate fields

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

## Unramified/totally ramified tower

 Unramified subfield: 11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of $$x^{4} + 8 x^{2} + 10 x + 2$$ x^4 + 8*x^2 + 10*x + 2 Relative Eisenstein polynomial: $$x^{3} + 11 t^{2}$$ x^3 + 11*t^2 $\ \in\Q_{11}(t)[x]$

## Ramification polygon

 Residual polynomials: $z^{2} + 3z + 3$ Associated inertia: $1$ Indices of inseparability: $[0]$

## Invariants of the Galois closure

 Galois group: $C_3:C_{12}$ (as 12T19) Inertia group: Intransitive group isomorphic to $C_3$ Wild inertia group: $C_1$ Unramified degree: $12$ Tame degree: $3$ Wild slopes: None Galois mean slope: $2/3$ Galois splitting model: $x^{12} - 3 x^{11} - 112 x^{10} + 135 x^{9} + 4708 x^{8} + 2405 x^{7} - 82903 x^{6} - 159417 x^{5} + 429601 x^{4} + 1548270 x^{3} + 1263093 x^{2} + 15394 x - 41348$ x^12 - 3*x^11 - 112*x^10 + 135*x^9 + 4708*x^8 + 2405*x^7 - 82903*x^6 - 159417*x^5 + 429601*x^4 + 1548270*x^3 + 1263093*x^2 + 15394*x - 41348