# Properties

 Label 12.385...144.18t315.a.a Dimension $12$ Group $S_3\wr S_3$ Conductor $3.851\times 10^{17}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$385129317228262144$$$$\medspace = 2^{8} \cdot 19^{5} \cdot 157^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.1309465408.1 Galois orbit size: $1$ Smallest permutation container: 18T315 Parity: odd Determinant: 1.19.2t1.a.a Projective image: $S_3\wr S_3$ Projective stem field: Galois closure of 9.1.1309465408.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - x^{8} + x^{7} + 2x^{5} - 5x^{4} + 5x^{3} - x^{2} - 2x + 1$$ x^9 - x^8 + x^7 + 2*x^5 - 5*x^4 + 5*x^3 - x^2 - 2*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{3} + 2x + 9$$

Roots:
 $r_{ 1 }$ $=$ $$2 a^{2} + a + 9 + 8 a\cdot 11 + \left(4 a^{2} + 7 a + 2\right)\cdot 11^{2} + \left(3 a^{2} + 3 a\right)\cdot 11^{3} + \left(8 a^{2} + 7 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 8 a + 2\right)\cdot 11^{5} + \left(a + 7\right)\cdot 11^{6} + \left(8 a^{2} + a + 1\right)\cdot 11^{7} + \left(4 a^{2} + 8 a + 8\right)\cdot 11^{8} + \left(8 a^{2} + 7 a + 6\right)\cdot 11^{9} +O(11^{10})$$ 2*a^2 + a + 9 + 8*a*11 + (4*a^2 + 7*a + 2)*11^2 + (3*a^2 + 3*a)*11^3 + (8*a^2 + 7*a + 1)*11^4 + (9*a^2 + 8*a + 2)*11^5 + (a + 7)*11^6 + (8*a^2 + a + 1)*11^7 + (4*a^2 + 8*a + 8)*11^8 + (8*a^2 + 7*a + 6)*11^9+O(11^10) $r_{ 2 }$ $=$ $$10 a^{2} + 7 a + 4 + \left(5 a^{2} + 2 a + 6\right)\cdot 11 + \left(2 a^{2} + 5 a + 4\right)\cdot 11^{2} + \left(8 a^{2} + 7 a + 6\right)\cdot 11^{3} + \left(3 a^{2} + 1\right)\cdot 11^{4} + \left(4 a^{2} + 3 a\right)\cdot 11^{5} + \left(7 a^{2} + 5 a\right)\cdot 11^{6} + \left(a^{2} + 3\right)\cdot 11^{7} + \left(10 a^{2} + 7 a + 10\right)\cdot 11^{8} + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{9} +O(11^{10})$$ 10*a^2 + 7*a + 4 + (5*a^2 + 2*a + 6)*11 + (2*a^2 + 5*a + 4)*11^2 + (8*a^2 + 7*a + 6)*11^3 + (3*a^2 + 1)*11^4 + (4*a^2 + 3*a)*11^5 + (7*a^2 + 5*a)*11^6 + (a^2 + 3)*11^7 + (10*a^2 + 7*a + 10)*11^8 + (10*a^2 + 9*a + 7)*11^9+O(11^10) $r_{ 3 }$ $=$ $$7 a^{2} + 6 a + 9 + \left(4 a^{2} + 8 a + 10\right)\cdot 11 + \left(2 a^{2} + 10 a + 8\right)\cdot 11^{2} + \left(2 a^{2} + 5 a\right)\cdot 11^{3} + \left(2 a^{2} + 6 a + 9\right)\cdot 11^{4} + \left(2 a^{2} + 5 a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 6 a + 1\right)\cdot 11^{6} + \left(4 a^{2} + 3 a\right)\cdot 11^{7} + \left(8 a^{2} + a + 9\right)\cdot 11^{8} + \left(3 a^{2} + 7 a + 8\right)\cdot 11^{9} +O(11^{10})$$ 7*a^2 + 6*a + 9 + (4*a^2 + 8*a + 10)*11 + (2*a^2 + 10*a + 8)*11^2 + (2*a^2 + 5*a)*11^3 + (2*a^2 + 6*a + 9)*11^4 + (2*a^2 + 5*a + 4)*11^5 + (9*a^2 + 6*a + 1)*11^6 + (4*a^2 + 3*a)*11^7 + (8*a^2 + a + 9)*11^8 + (3*a^2 + 7*a + 8)*11^9+O(11^10) $r_{ 4 }$ $=$ $$a^{2} + 4 a + 4 + \left(2 a^{2} + 4 a + 3\right)\cdot 11 + \left(2 a^{2} + 2 a + 3\right)\cdot 11^{2} + \left(10 a^{2} + 5 a + 9\right)\cdot 11^{3} + \left(2 a + 5\right)\cdot 11^{4} + \left(5 a^{2} + 10\right)\cdot 11^{5} + \left(2 a^{2} + 8 a + 1\right)\cdot 11^{6} + \left(3 a^{2} + 5 a + 6\right)\cdot 11^{7} + \left(10 a^{2} + 4\right)\cdot 11^{8} + \left(5 a^{2} + 7 a + 3\right)\cdot 11^{9} +O(11^{10})$$ a^2 + 4*a + 4 + (2*a^2 + 4*a + 3)*11 + (2*a^2 + 2*a + 3)*11^2 + (10*a^2 + 5*a + 9)*11^3 + (2*a + 5)*11^4 + (5*a^2 + 10)*11^5 + (2*a^2 + 8*a + 1)*11^6 + (3*a^2 + 5*a + 6)*11^7 + (10*a^2 + 4)*11^8 + (5*a^2 + 7*a + 3)*11^9+O(11^10) $r_{ 5 }$ $=$ $$9 a^{2} + 10 a + 10 + \left(a^{2} + 7 a\right)\cdot 11 + \left(7 a^{2} + 7 a + 7\right)\cdot 11^{2} + \left(9 a^{2} + 5 a + 4\right)\cdot 11^{3} + \left(3 a^{2} + 7 a + 5\right)\cdot 11^{4} + \left(10 a^{2} + 2 a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 10 a + 3\right)\cdot 11^{6} + \left(3 a^{2} + 8 a + 2\right)\cdot 11^{7} + \left(5 a^{2} + 7 a\right)\cdot 11^{8} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{9} +O(11^{10})$$ 9*a^2 + 10*a + 10 + (a^2 + 7*a)*11 + (7*a^2 + 7*a + 7)*11^2 + (9*a^2 + 5*a + 4)*11^3 + (3*a^2 + 7*a + 5)*11^4 + (10*a^2 + 2*a + 4)*11^5 + (9*a^2 + 10*a + 3)*11^6 + (3*a^2 + 8*a + 2)*11^7 + (5*a^2 + 7*a)*11^8 + (5*a^2 + 9*a + 8)*11^9+O(11^10) $r_{ 6 }$ $=$ $$3 a^{2} + 5 a + 2 + \left(3 a^{2} + 10\right)\cdot 11 + \left(a^{2} + 9 a + 2\right)\cdot 11^{2} + \left(4 a^{2} + 8 a + 8\right)\cdot 11^{3} + \left(3 a^{2} + 2 a + 4\right)\cdot 11^{4} + \left(7 a^{2} + 5 a\right)\cdot 11^{5} + \left(4 a^{2} + 6 a\right)\cdot 11^{6} + \left(5 a^{2} + a + 8\right)\cdot 11^{7} + \left(6 a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(5 a^{2} + 2 a + 8\right)\cdot 11^{9} +O(11^{10})$$ 3*a^2 + 5*a + 2 + (3*a^2 + 10)*11 + (a^2 + 9*a + 2)*11^2 + (4*a^2 + 8*a + 8)*11^3 + (3*a^2 + 2*a + 4)*11^4 + (7*a^2 + 5*a)*11^5 + (4*a^2 + 6*a)*11^6 + (5*a^2 + a + 8)*11^7 + (6*a^2 + 7*a + 1)*11^8 + (5*a^2 + 2*a + 8)*11^9+O(11^10) $r_{ 7 }$ $=$ $$a^{2} + 9 a + 1 + \left(7 a^{2} + 3 a + 3\right)\cdot 11 + \left(5 a^{2} + 6 a + 2\right)\cdot 11^{2} + \left(10 a^{2} + 6 a + 8\right)\cdot 11^{3} + \left(2 a^{2} + 2 a + 2\right)\cdot 11^{4} + \left(5 a^{2} + 6 a + 5\right)\cdot 11^{5} + \left(10 a^{2} + 9 a + 10\right)\cdot 11^{6} + \left(7 a^{2} + 9 a + 7\right)\cdot 11^{7} + \left(4 a^{2} + 9 a + 7\right)\cdot 11^{8} + \left(a^{2} + 5\right)\cdot 11^{9} +O(11^{10})$$ a^2 + 9*a + 1 + (7*a^2 + 3*a + 3)*11 + (5*a^2 + 6*a + 2)*11^2 + (10*a^2 + 6*a + 8)*11^3 + (2*a^2 + 2*a + 2)*11^4 + (5*a^2 + 6*a + 5)*11^5 + (10*a^2 + 9*a + 10)*11^6 + (7*a^2 + 9*a + 7)*11^7 + (4*a^2 + 9*a + 7)*11^8 + (a^2 + 5)*11^9+O(11^10) $r_{ 8 }$ $=$ $$3 a^{2} + 7 a + \left(10 a^{2} + 9 a\right)\cdot 11 + \left(2 a^{2} + 4 a + 6\right)\cdot 11^{2} + \left(9 a^{2} + 9 a + 6\right)\cdot 11^{3} + \left(5 a^{2} + a + 6\right)\cdot 11^{4} + \left(3 a^{2} + 10 a + 6\right)\cdot 11^{5} + \left(2 a^{2} + 5 a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 8 a + 5\right)\cdot 11^{7} + \left(8 a^{2} + 10 a + 9\right)\cdot 11^{8} + \left(5 a^{2} + 2 a + 7\right)\cdot 11^{9} +O(11^{10})$$ 3*a^2 + 7*a + (10*a^2 + 9*a)*11 + (2*a^2 + 4*a + 6)*11^2 + (9*a^2 + 9*a + 6)*11^3 + (5*a^2 + a + 6)*11^4 + (3*a^2 + 10*a + 6)*11^5 + (2*a^2 + 5*a + 10)*11^6 + (9*a^2 + 8*a + 5)*11^7 + (8*a^2 + 10*a + 9)*11^8 + (5*a^2 + 2*a + 7)*11^9+O(11^10) $r_{ 9 }$ $=$ $$8 a^{2} + 6 a + 6 + \left(8 a^{2} + 9 a + 8\right)\cdot 11 + \left(4 a^{2} + 6\right)\cdot 11^{2} + \left(8 a^{2} + 2 a + 10\right)\cdot 11^{3} + \left(a^{2} + a + 6\right)\cdot 11^{4} + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{5} + \left(7 a^{2} + a + 8\right)\cdot 11^{6} + \left(10 a^{2} + 4 a + 8\right)\cdot 11^{7} + \left(6 a^{2} + 2 a + 3\right)\cdot 11^{8} + \left(7 a^{2} + 7 a + 9\right)\cdot 11^{9} +O(11^{10})$$ 8*a^2 + 6*a + 6 + (8*a^2 + 9*a + 8)*11 + (4*a^2 + 6)*11^2 + (8*a^2 + 2*a + 10)*11^3 + (a^2 + a + 6)*11^4 + (7*a^2 + 2*a + 9)*11^5 + (7*a^2 + a + 8)*11^6 + (10*a^2 + 4*a + 8)*11^7 + (6*a^2 + 2*a + 3)*11^8 + (7*a^2 + 7*a + 9)*11^9+O(11^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(3,4)$ $(3,4,6)$ $(1,3,5)(2,4,7)(6,9,8)$ $(3,5)(4,7)(6,9)$ $(1,2,8)$ $(5,7,9)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $12$ $9$ $2$ $(3,4)$ $4$ $18$ $2$ $(3,5)(4,7)(6,9)$ $2$ $27$ $2$ $(1,2)(3,4)(5,7)$ $0$ $27$ $2$ $(1,2)(3,4)$ $0$ $54$ $2$ $(1,3)(2,4)(5,7)(6,8)$ $2$ $6$ $3$ $(1,2,8)$ $0$ $8$ $3$ $(1,8,2)(3,6,4)(5,9,7)$ $3$ $12$ $3$ $(1,8,2)(3,6,4)$ $-3$ $72$ $3$ $(1,3,5)(2,4,7)(6,9,8)$ $0$ $54$ $4$ $(1,4,2,3)(6,8)$ $0$ $162$ $4$ $(1,4,2,3)(5,7)(6,8)$ $0$ $36$ $6$ $(1,2,8)(3,5)(4,7)(6,9)$ $2$ $36$ $6$ $(1,6,8,4,2,3)$ $-1$ $36$ $6$ $(1,2,8)(3,4)$ $-2$ $36$ $6$ $(1,2,8)(3,4)(5,7,9)$ $1$ $54$ $6$ $(1,8,2)(3,4)(5,7)$ $0$ $72$ $6$ $(1,2,8)(3,5,6,9,4,7)$ $-1$ $108$ $6$ $(1,6,8,4,2,3)(5,7)$ $-1$ $216$ $6$ $(1,4,7,2,3,5)(6,9,8)$ $0$ $144$ $9$ $(1,6,9,8,4,7,2,3,5)$ $0$ $108$ $12$ $(1,2,8)(3,5,4,7)(6,9)$ $0$

The blue line marks the conjugacy class containing complex conjugation.