# Properties

 Label 8.336...875.18t157.a.a Dimension $8$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $3.370\times 10^{15}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $8$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$3369740538796875$$$$\medspace = 3^{5} \cdot 5^{6} \cdot 31^{6}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.374415615421875.1 Galois orbit size: $1$ Smallest permutation container: 18T157 Parity: odd Determinant: 1.3.2t1.a.a Projective image: $C_3^2:\GL(2,3)$ Projective stem field: Galois closure of 9.3.374415615421875.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{8} - 9x^{7} - 4x^{6} + 90x^{5} - 14x^{4} - 192x^{3} - 55x^{2} + 113x + 46$$ x^9 - 3*x^8 - 9*x^7 - 4*x^6 + 90*x^5 - 14*x^4 - 192*x^3 - 55*x^2 + 113*x + 46 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{4} + 3x^{2} + 12x + 2$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.