# Properties

 Label 8.127745014464.9t26.a.a Dimension $8$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $127745014464$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $8$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$127745014464$$$$\medspace = 2^{6} \cdot 3^{7} \cdot 97^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.127745014464.2 Galois orbit size: $1$ Smallest permutation container: $((C_3^2:Q_8):C_3):C_2$ Parity: odd Determinant: 1.291.2t1.a.a Projective image: $C_3^2:\GL(2,3)$ Projective stem field: Galois closure of 9.3.127745014464.2

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - x^{8} - 7x^{7} + 8x^{6} + 8x^{5} - x^{4} - 25x^{3} + 11x^{2} + 8x + 1$$ x^9 - x^8 - 7*x^7 + 8*x^6 + 8*x^5 - x^4 - 25*x^3 + 11*x^2 + 8*x + 1 .

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.