# Properties

 Label 12.201...875.18t218.a.a Dimension $12$ Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor $2.018\times 10^{17}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor: $$201785586220296875$$$$\medspace = 5^{6} \cdot 419^{5}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.770541618025.1 Galois orbit size: $1$ Smallest permutation container: 18T218 Parity: odd Determinant: 1.419.2t1.a.a Projective image: $C_3^3:S_4$ Projective stem field: Galois closure of 9.1.770541618025.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} + x^{7} - 8x^{6} - 5x^{5} - 2x^{4} + 20x^{3} + 5x^{2} + 23x + 5$$ x^9 + x^7 - 8*x^6 - 5*x^5 - 2*x^4 + 20*x^3 + 5*x^2 + 23*x + 5 .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{3} + 2x + 18$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.