# Properties

 Label 16.166...936.24t1334.b Dimension $16$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $1.669\times 10^{26}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $16$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$166\!\cdots\!936$$$$\medspace = 2^{10} \cdot 3^{18} \cdot 29^{10}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.3.749306075343552.2 Galois orbit size: $1$ Smallest permutation container: 24T1334 Parity: even Projective image: $C_3^2:\GL(2,3)$ Projective field: Galois closure of 9.3.749306075343552.2

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character values $c1$ $1$ $1$ $()$ $16$ $9$ $2$ $(1,5)(2,9)(3,8)(4,7)$ $0$ $36$ $2$ $(3,4)(5,8)(6,9)$ $0$ $8$ $3$ $(1,8,9)(2,3,5)(4,7,6)$ $-2$ $24$ $3$ $(1,5,6)(3,9,7)$ $-2$ $48$ $3$ $(1,2,3)(4,7,5)(6,8,9)$ $1$ $54$ $4$ $(1,4,5,7)(2,3,9,8)$ $0$ $72$ $6$ $(1,4,3)(2,5,6,7,9,8)$ $0$ $72$ $6$ $(1,3,2,8,7,6)(4,5)$ $0$ $54$ $8$ $(1,2,4,3,5,9,7,8)$ $0$ $54$ $8$ $(1,9,4,8,5,2,7,3)$ $0$
The blue line marks the conjugacy class containing complex conjugation.