# Properties

 Label 16.182...281.24t1334.a.a Dimension $16$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $1.827\times 10^{26}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $16$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$182\!\cdots\!281$$$$\medspace = 3^{28} \cdot 41^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.988941019347.2 Galois orbit size: $1$ Smallest permutation container: 24T1334 Parity: even Determinant: 1.1.1t1.a.a Projective image: $\AGL(2,3)$ Projective stem field: Galois closure of 9.3.988941019347.2

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{8} + 3x^{7} - 6x^{6} + 12x^{5} - 21x^{4} + 39x^{3} - 39x^{2} + 3x + 10$$ x^9 - 3*x^8 + 3*x^7 - 6*x^6 + 12*x^5 - 21*x^4 + 39*x^3 - 39*x^2 + 3*x + 10 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{4} + 7x^{2} + 10x + 3$$

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

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $16$ $9$ $2$ $(1,8)(3,9)(4,6)(5,7)$ $0$ $36$ $2$ $(1,2)(3,5)(4,7)$ $0$ $8$ $3$ $(1,3,4)(2,5,7)(6,9,8)$ $-2$ $24$ $3$ $(1,6,7)(4,5,8)$ $-2$ $48$ $3$ $(1,5,6)(2,8,4)(3,7,9)$ $1$ $54$ $4$ $(1,9,8,3)(4,5,6,7)$ $0$ $72$ $6$ $(1,2,6,9,7,3)(4,8)$ $0$ $72$ $6$ $(1,5,4,2,3,7)(6,9,8)$ $0$ $54$ $8$ $(1,2,9,8,3,6,7,5)$ $0$ $54$ $8$ $(1,6,9,5,3,2,7,8)$ $0$

The blue line marks the conjugacy class containing complex conjugation.