# Properties

 Label 7.15.10.1 Base $$\Q_{7}$$ Degree $$15$$ e $$3$$ f $$5$$ c $$10$$ Galois group $C_{15}$ (as 15T1)

# Related objects

## Defining polynomial

 $$x^{15} + 35 x^{12} + 3 x^{11} + 12 x^{10} + 490 x^{9} - 315 x^{8} - 2517 x^{7} + 3454 x^{6} - 834 x^{5} + 26565 x^{4} + 12846 x^{3} + 13662 x^{2} - 19944 x + 16290$$ x^15 + 35*x^12 + 3*x^11 + 12*x^10 + 490*x^9 - 315*x^8 - 2517*x^7 + 3454*x^6 - 834*x^5 + 26565*x^4 + 12846*x^3 + 13662*x^2 - 19944*x + 16290

## Invariants

 Base field: $\Q_{7}$ Degree $d$: $15$ Ramification exponent $e$: $3$ Residue field degree $f$: $5$ Discriminant exponent $c$: $10$ Discriminant root field: $\Q_{7}$ Root number: $1$ $\card{ \Gal(K/\Q_{ 7 }) }$: $15$ This field is Galois and abelian over $\Q_{7}.$ 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: 7.5.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of $$x^{5} + x + 4$$ x^5 + x + 4 Relative Eisenstein polynomial: $$x^{3} + 7$$ x^3 + 7 $\ \in\Q_{7}(t)[x]$

## Ramification polygon Residual polynomials: $z^{2} + 3z + 3$ Associated inertia: $1$ Indices of inseparability: $$

## Invariants of the Galois closure

 Galois group: $C_{15}$ (as 15T1) Inertia group: Intransitive group isomorphic to $C_3$ Wild inertia group: $C_1$ Unramified degree: $5$ Tame degree: $3$ Wild slopes: None Galois mean slope: $2/3$ Galois splitting model: Not computed