# Properties

 Label 11.12.10.4 Base $$\Q_{11}$$ Degree $$12$$ e $$6$$ f $$2$$ c $$10$$ Galois group $C_3\times (C_3 : C_4)$ (as 12T19)

# Related objects

## Defining polynomial

 $$x^{12} - 704 x^{6} - 121$$ x^12 - 704*x^6 - 121

## Invariants

 Base field: $\Q_{11}$ Degree $d$: $12$ Ramification exponent $e$: $6$ Residue field degree $f$: $2$ Discriminant exponent $c$: $10$ 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: $\Q_{11}(\sqrt{2})$ $\cong \Q_{11}(t)$ where $t$ is a root of $$x^{2} + 7 x + 2$$ x^2 + 7*x + 2 Relative Eisenstein polynomial: $$x^{6} + 110 t + 33$$ x^6 + 110*t + 33 $\ \in\Q_{11}(t)[x]$

## Ramification polygon

 Residual polynomials: $z^{5} + 6z^{4} + 4z^{3} + 9z^{2} + 4z + 6$ 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 $C_6$ Wild inertia group: $C_1$ Unramified degree: $6$ Tame degree: $6$ Wild slopes: None Galois mean slope: $5/6$ Galois splitting model: $x^{12} - 3 x^{11} + 41 x^{10} - 273 x^{9} + 5320 x^{8} - 1216 x^{7} + 185714 x^{6} - 462051 x^{5} + 6328363 x^{4} + 19619049 x^{3} + 296778921 x^{2} + 695676814 x + 2813125384$ x^12 - 3*x^11 + 41*x^10 - 273*x^9 + 5320*x^8 - 1216*x^7 + 185714*x^6 - 462051*x^5 + 6328363*x^4 + 19619049*x^3 + 296778921*x^2 + 695676814*x + 2813125384