# Properties

 Label 12.331...609.18t206.a.a Dimension $12$ Group $S_3 \wr C_3$ Conductor $3.313\times 10^{17}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3 \wr C_3$ Conductor: $$331258496613663609$$$$\medspace = 3^{26} \cdot 19^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.5.22082967873.1 Galois orbit size: $1$ Smallest permutation container: 18T206 Parity: even Determinant: 1.1.1t1.a.a Projective image: $C_3^3:C_2^2.C_6$ Projective stem field: Galois closure of 9.5.22082967873.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 9x^{7} + 27x^{5} - 36x^{3} + 27x - 9$$ x^9 - 9*x^7 + 27*x^5 - 36*x^3 + 27*x - 9 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{3} + 2x + 27$$

Roots:
 $r_{ 1 }$ $=$ $$10 a^{2} + 8 a + 3 + \left(10 a^{2} + 5 a + 25\right)\cdot 29 + \left(15 a^{2} + 21\right)\cdot 29^{2} + \left(27 a^{2} + 22 a + 4\right)\cdot 29^{3} + \left(19 a^{2} + 18 a + 1\right)\cdot 29^{4} + \left(20 a^{2} + 27 a + 14\right)\cdot 29^{5} + \left(27 a^{2} + 28 a + 28\right)\cdot 29^{6} + \left(10 a^{2} + 11 a + 26\right)\cdot 29^{7} + \left(18 a^{2} + 6 a + 8\right)\cdot 29^{8} + \left(15 a^{2} + 20 a + 18\right)\cdot 29^{9} +O(29^{10})$$ 10*a^2 + 8*a + 3 + (10*a^2 + 5*a + 25)*29 + (15*a^2 + 21)*29^2 + (27*a^2 + 22*a + 4)*29^3 + (19*a^2 + 18*a + 1)*29^4 + (20*a^2 + 27*a + 14)*29^5 + (27*a^2 + 28*a + 28)*29^6 + (10*a^2 + 11*a + 26)*29^7 + (18*a^2 + 6*a + 8)*29^8 + (15*a^2 + 20*a + 18)*29^9+O(29^10) $r_{ 2 }$ $=$ $$23 a^{2} + 24 a + 3 + \left(23 a^{2} + 26 a + 9\right)\cdot 29 + \left(28 a^{2} + 4 a + 18\right)\cdot 29^{2} + \left(22 a^{2} + 28\right)\cdot 29^{3} + \left(2 a^{2} + 23 a + 8\right)\cdot 29^{4} + \left(13 a^{2} + a + 28\right)\cdot 29^{5} + \left(27 a^{2} + 7 a + 5\right)\cdot 29^{6} + \left(25 a^{2} + 11 a + 23\right)\cdot 29^{7} + \left(19 a^{2} + 8 a + 26\right)\cdot 29^{8} + \left(14 a^{2} + 22 a + 5\right)\cdot 29^{9} +O(29^{10})$$ 23*a^2 + 24*a + 3 + (23*a^2 + 26*a + 9)*29 + (28*a^2 + 4*a + 18)*29^2 + (22*a^2 + 28)*29^3 + (2*a^2 + 23*a + 8)*29^4 + (13*a^2 + a + 28)*29^5 + (27*a^2 + 7*a + 5)*29^6 + (25*a^2 + 11*a + 23)*29^7 + (19*a^2 + 8*a + 26)*29^8 + (14*a^2 + 22*a + 5)*29^9+O(29^10) $r_{ 3 }$ $=$ $$7 a^{2} + 22 a + 28 + \left(4 a^{2} + 16\right)\cdot 29 + \left(7 a^{2} + 25 a + 20\right)\cdot 29^{2} + \left(14 a^{2} + 24 a + 25\right)\cdot 29^{3} + \left(3 a^{2} + 16 a + 17\right)\cdot 29^{4} + \left(27 a^{2} + 27 a + 22\right)\cdot 29^{5} + \left(18 a^{2} + 23 a + 16\right)\cdot 29^{6} + \left(8 a^{2} + 8 a + 4\right)\cdot 29^{7} + \left(10 a^{2} + 7 a + 27\right)\cdot 29^{8} + \left(10 a^{2} + 22 a + 20\right)\cdot 29^{9} +O(29^{10})$$ 7*a^2 + 22*a + 28 + (4*a^2 + 16)*29 + (7*a^2 + 25*a + 20)*29^2 + (14*a^2 + 24*a + 25)*29^3 + (3*a^2 + 16*a + 17)*29^4 + (27*a^2 + 27*a + 22)*29^5 + (18*a^2 + 23*a + 16)*29^6 + (8*a^2 + 8*a + 4)*29^7 + (10*a^2 + 7*a + 27)*29^8 + (10*a^2 + 22*a + 20)*29^9+O(29^10) $r_{ 4 }$ $=$ $$12 a^{2} + 28 a + 25 + \left(14 a^{2} + 22 a + 20\right)\cdot 29 + \left(6 a^{2} + 3 a + 19\right)\cdot 29^{2} + \left(16 a^{2} + 11 a + 18\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 20\right)\cdot 29^{4} + \left(10 a^{2} + 2 a + 9\right)\cdot 29^{5} + \left(11 a^{2} + 5 a + 16\right)\cdot 29^{6} + \left(9 a^{2} + 8 a + 5\right)\cdot 29^{7} + \left(15 a + 4\right)\cdot 29^{8} + \left(3 a^{2} + 15 a + 11\right)\cdot 29^{9} +O(29^{10})$$ 12*a^2 + 28*a + 25 + (14*a^2 + 22*a + 20)*29 + (6*a^2 + 3*a + 19)*29^2 + (16*a^2 + 11*a + 18)*29^3 + (5*a^2 + 22*a + 20)*29^4 + (10*a^2 + 2*a + 9)*29^5 + (11*a^2 + 5*a + 16)*29^6 + (9*a^2 + 8*a + 5)*29^7 + (15*a + 4)*29^8 + (3*a^2 + 15*a + 11)*29^9+O(29^10) $r_{ 5 }$ $=$ $$21 a^{2} + 16 a + 10 + \left(7 a^{2} + 27 a + 26\right)\cdot 29 + \left(21 a^{2} + 11 a + 17\right)\cdot 29^{2} + \left(21 a^{2} + 15 a + 7\right)\cdot 29^{3} + \left(28 a^{2} + 5 a + 24\right)\cdot 29^{4} + \left(24 a^{2} + 26 a + 24\right)\cdot 29^{5} + \left(a + 18\right)\cdot 29^{6} + \left(9 a + 17\right)\cdot 29^{7} + \left(13 a^{2} + 8 a + 17\right)\cdot 29^{8} + \left(14 a^{2} + 21 a + 5\right)\cdot 29^{9} +O(29^{10})$$ 21*a^2 + 16*a + 10 + (7*a^2 + 27*a + 26)*29 + (21*a^2 + 11*a + 17)*29^2 + (21*a^2 + 15*a + 7)*29^3 + (28*a^2 + 5*a + 24)*29^4 + (24*a^2 + 26*a + 24)*29^5 + (a + 18)*29^6 + (9*a + 17)*29^7 + (13*a^2 + 8*a + 17)*29^8 + (14*a^2 + 21*a + 5)*29^9+O(29^10) $r_{ 6 }$ $=$ $$5 a^{2} + 3 a + 6 + \left(21 a^{2} + 20 a + 20\right)\cdot 29 + \left(5 a^{2} + 16 a + 16\right)\cdot 29^{2} + \left(17 a^{2} + 22 a + 8\right)\cdot 29^{3} + \left(11 a^{2} + 9 a + 16\right)\cdot 29^{4} + \left(17 a^{2} + 6\right)\cdot 29^{5} + \left(9 a + 1\right)\cdot 29^{6} + \left(15 a^{2} + 8 a + 19\right)\cdot 29^{7} + \left(14 a^{2} + 10 a + 5\right)\cdot 29^{8} + \left(13 a^{2} + 23 a + 5\right)\cdot 29^{9} +O(29^{10})$$ 5*a^2 + 3*a + 6 + (21*a^2 + 20*a + 20)*29 + (5*a^2 + 16*a + 16)*29^2 + (17*a^2 + 22*a + 8)*29^3 + (11*a^2 + 9*a + 16)*29^4 + (17*a^2 + 6)*29^5 + (9*a + 1)*29^6 + (15*a^2 + 8*a + 19)*29^7 + (14*a^2 + 10*a + 5)*29^8 + (13*a^2 + 23*a + 5)*29^9+O(29^10) $r_{ 7 }$ $=$ $$25 a^{2} + 14 a + 23 + \left(6 a^{2} + 7 a + 10\right)\cdot 29 + \left(a^{2} + 13 a + 20\right)\cdot 29^{2} + \left(20 a^{2} + 2 a + 2\right)\cdot 29^{3} + \left(23 a^{2} + a + 13\right)\cdot 29^{4} + \left(22 a^{2} + 23\right)\cdot 29^{5} + \left(16 a^{2} + 22 a + 22\right)\cdot 29^{6} + \left(19 a^{2} + 11 a + 5\right)\cdot 29^{7} + \left(15 a^{2} + 5 a + 7\right)\cdot 29^{8} + \left(11 a^{2} + 21 a + 12\right)\cdot 29^{9} +O(29^{10})$$ 25*a^2 + 14*a + 23 + (6*a^2 + 7*a + 10)*29 + (a^2 + 13*a + 20)*29^2 + (20*a^2 + 2*a + 2)*29^3 + (23*a^2 + a + 13)*29^4 + (22*a^2 + 23)*29^5 + (16*a^2 + 22*a + 22)*29^6 + (19*a^2 + 11*a + 5)*29^7 + (15*a^2 + 5*a + 7)*29^8 + (11*a^2 + 21*a + 12)*29^9+O(29^10) $r_{ 8 }$ $=$ $$28 a^{2} + 12 a + 27 + \left(a + 2\right)\cdot 29 + \left(22 a^{2} + 28 a + 19\right)\cdot 29^{2} + \left(20 a^{2} + 3 a + 3\right)\cdot 29^{3} + \left(22 a^{2} + 18 a + 2\right)\cdot 29^{4} + \left(17 a^{2} + 28 a + 7\right)\cdot 29^{5} + \left(11 a^{2} + 26 a + 6\right)\cdot 29^{6} + \left(23 a^{2} + 8 a + 1\right)\cdot 29^{7} + \left(27 a^{2} + 13 a + 4\right)\cdot 29^{8} + \left(3 a^{2} + 13 a + 2\right)\cdot 29^{9} +O(29^{10})$$ 28*a^2 + 12*a + 27 + (a + 2)*29 + (22*a^2 + 28*a + 19)*29^2 + (20*a^2 + 3*a + 3)*29^3 + (22*a^2 + 18*a + 2)*29^4 + (17*a^2 + 28*a + 7)*29^5 + (11*a^2 + 26*a + 6)*29^6 + (23*a^2 + 8*a + 1)*29^7 + (27*a^2 + 13*a + 4)*29^8 + (3*a^2 + 13*a + 2)*29^9+O(29^10) $r_{ 9 }$ $=$ $$14 a^{2} + 18 a + 20 + \left(26 a^{2} + 3 a + 12\right)\cdot 29 + \left(7 a^{2} + 12 a + 19\right)\cdot 29^{2} + \left(13 a^{2} + 13 a + 15\right)\cdot 29^{3} + \left(26 a^{2} + 11\right)\cdot 29^{4} + \left(19 a^{2} + a + 8\right)\cdot 29^{5} + \left(20 a + 28\right)\cdot 29^{6} + \left(3 a^{2} + 8 a + 11\right)\cdot 29^{7} + \left(25 a^{2} + 12 a + 14\right)\cdot 29^{8} + \left(28 a^{2} + 14 a + 5\right)\cdot 29^{9} +O(29^{10})$$ 14*a^2 + 18*a + 20 + (26*a^2 + 3*a + 12)*29 + (7*a^2 + 12*a + 19)*29^2 + (13*a^2 + 13*a + 15)*29^3 + (26*a^2 + 11)*29^4 + (19*a^2 + a + 8)*29^5 + (20*a + 28)*29^6 + (3*a^2 + 8*a + 11)*29^7 + (25*a^2 + 12*a + 14)*29^8 + (28*a^2 + 14*a + 5)*29^9+O(29^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.