# Properties

 Label 8.282...203.18t157.a Dimension $8$ Group $((C_3^2:Q_8):C_3):C_2$ Conductor $2.824\times 10^{12}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $8$ Group: $((C_3^2:Q_8):C_3):C_2$ Conductor: $$2824440448203$$$$\medspace = 3^{13} \cdot 11^{6}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.3.2824440448203.2 Galois orbit size: $1$ Smallest permutation container: 18T157 Parity: odd Projective image: $C_3^2:\GL(2,3)$ Projective field: Galois closure of 9.3.2824440448203.2

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{4} + 2x^{2} + 11x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$4 a^{3} + 7 a^{2} + a + 16 + \left(4 a^{3} + 6 a^{2} + 2 a + 17\right)\cdot 19 + \left(7 a^{3} + 12 a^{2} + 3 a + 18\right)\cdot 19^{2} + \left(2 a^{3} + 9 a + 8\right)\cdot 19^{3} + \left(a^{3} + 5 a^{2} + 14 a + 6\right)\cdot 19^{4} + \left(17 a^{3} + 17 a^{2} + 7 a + 12\right)\cdot 19^{5} + \left(5 a^{3} + 18 a^{2} + 10 a + 13\right)\cdot 19^{6} + \left(17 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 19^{7} + \left(3 a^{3} + 9 a^{2} + 11 a + 4\right)\cdot 19^{8} + \left(16 a^{3} + 18 a^{2} + 15 a + 10\right)\cdot 19^{9} +O(19^{10})$$ 4*a^3 + 7*a^2 + a + 16 + (4*a^3 + 6*a^2 + 2*a + 17)*19 + (7*a^3 + 12*a^2 + 3*a + 18)*19^2 + (2*a^3 + 9*a + 8)*19^3 + (a^3 + 5*a^2 + 14*a + 6)*19^4 + (17*a^3 + 17*a^2 + 7*a + 12)*19^5 + (5*a^3 + 18*a^2 + 10*a + 13)*19^6 + (17*a^3 + 17*a^2 + 14*a + 4)*19^7 + (3*a^3 + 9*a^2 + 11*a + 4)*19^8 + (16*a^3 + 18*a^2 + 15*a + 10)*19^9+O(19^10) $r_{ 2 }$ $=$ $$9 a^{3} + 13 a^{2} + 11 a + 11 + \left(15 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 19 + \left(7 a^{3} + 13 a^{2} + 7 a + 15\right)\cdot 19^{2} + \left(9 a^{3} + 3 a^{2} + 7 a + 17\right)\cdot 19^{3} + \left(10 a^{3} + 16 a^{2} + 12 a + 18\right)\cdot 19^{4} + \left(10 a^{3} + 11 a^{2} + 4 a + 14\right)\cdot 19^{5} + \left(16 a^{3} + 15 a^{2} + 3 a + 12\right)\cdot 19^{6} + \left(14 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 19^{7} + \left(18 a^{3} + 12 a^{2} + 3 a + 15\right)\cdot 19^{8} + \left(10 a^{3} + 18 a^{2} + a + 14\right)\cdot 19^{9} +O(19^{10})$$ 9*a^3 + 13*a^2 + 11*a + 11 + (15*a^3 + 2*a^2 + 7*a + 2)*19 + (7*a^3 + 13*a^2 + 7*a + 15)*19^2 + (9*a^3 + 3*a^2 + 7*a + 17)*19^3 + (10*a^3 + 16*a^2 + 12*a + 18)*19^4 + (10*a^3 + 11*a^2 + 4*a + 14)*19^5 + (16*a^3 + 15*a^2 + 3*a + 12)*19^6 + (14*a^3 + 6*a^2 + 12*a + 6)*19^7 + (18*a^3 + 12*a^2 + 3*a + 15)*19^8 + (10*a^3 + 18*a^2 + a + 14)*19^9+O(19^10) $r_{ 3 }$ $=$ $$17 a^{3} + 15 a^{2} + 9 a + 3 + \left(10 a^{3} + 7 a^{2} + 18 a + 3\right)\cdot 19 + \left(18 a^{3} + 10 a^{2} + 3 a + 6\right)\cdot 19^{2} + \left(18 a^{3} + 7 a^{2} + 11 a + 10\right)\cdot 19^{3} + \left(18 a^{3} + 8 a^{2} + 8 a\right)\cdot 19^{4} + \left(9 a^{3} + 4 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(12 a^{3} + 16 a^{2} + 14 a + 18\right)\cdot 19^{6} + \left(13 a^{3} + 11 a^{2} + 14 a + 15\right)\cdot 19^{7} + \left(17 a^{3} + 10 a^{2} + 10 a + 4\right)\cdot 19^{8} + \left(10 a^{3} + 8 a^{2} + 16 a + 4\right)\cdot 19^{9} +O(19^{10})$$ 17*a^3 + 15*a^2 + 9*a + 3 + (10*a^3 + 7*a^2 + 18*a + 3)*19 + (18*a^3 + 10*a^2 + 3*a + 6)*19^2 + (18*a^3 + 7*a^2 + 11*a + 10)*19^3 + (18*a^3 + 8*a^2 + 8*a)*19^4 + (9*a^3 + 4*a^2 + 3*a + 3)*19^5 + (12*a^3 + 16*a^2 + 14*a + 18)*19^6 + (13*a^3 + 11*a^2 + 14*a + 15)*19^7 + (17*a^3 + 10*a^2 + 10*a + 4)*19^8 + (10*a^3 + 8*a^2 + 16*a + 4)*19^9+O(19^10) $r_{ 4 }$ $=$ $$17 + 4\cdot 19 + 7\cdot 19^{2} + 17\cdot 19^{4} + 14\cdot 19^{5} + 13\cdot 19^{6} + 7\cdot 19^{7} + 11\cdot 19^{8} + 15\cdot 19^{9} +O(19^{10})$$ 17 + 4*19 + 7*19^2 + 17*19^4 + 14*19^5 + 13*19^6 + 7*19^7 + 11*19^8 + 15*19^9+O(19^10) $r_{ 5 }$ $=$ $$2 a^{3} + 2 a^{2} + a + 5 + \left(6 a^{3} + 8 a^{2} + 14 a + 9\right)\cdot 19 + \left(16 a^{3} + 6 a^{2} + 2 a + 16\right)\cdot 19^{2} + \left(10 a^{3} + 12 a^{2} + 12 a + 3\right)\cdot 19^{3} + \left(15 a^{3} + 18 a^{2} + 4 a + 8\right)\cdot 19^{4} + \left(4 a^{3} + 16 a^{2} + 9 a + 13\right)\cdot 19^{5} + \left(7 a^{3} + 8 a + 6\right)\cdot 19^{6} + \left(17 a^{3} + 5 a^{2} + 4 a + 3\right)\cdot 19^{7} + \left(7 a^{3} + 5 a + 15\right)\cdot 19^{8} + \left(11 a^{3} + 15 a^{2} + 4 a + 9\right)\cdot 19^{9} +O(19^{10})$$ 2*a^3 + 2*a^2 + a + 5 + (6*a^3 + 8*a^2 + 14*a + 9)*19 + (16*a^3 + 6*a^2 + 2*a + 16)*19^2 + (10*a^3 + 12*a^2 + 12*a + 3)*19^3 + (15*a^3 + 18*a^2 + 4*a + 8)*19^4 + (4*a^3 + 16*a^2 + 9*a + 13)*19^5 + (7*a^3 + 8*a + 6)*19^6 + (17*a^3 + 5*a^2 + 4*a + 3)*19^7 + (7*a^3 + 5*a + 15)*19^8 + (11*a^3 + 15*a^2 + 4*a + 9)*19^9+O(19^10) $r_{ 6 }$ $=$ $$2 a^{3} + 13 a^{2} + 15 a + 16 + \left(2 a^{3} + 11 a^{2} + 17 a + 17\right)\cdot 19 + \left(8 a^{3} + 13 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(17 a^{3} + 16 a^{2} + 10 a\right)\cdot 19^{3} + \left(12 a^{3} + a^{2} + 3 a + 17\right)\cdot 19^{4} + \left(13 a^{3} + 15 a^{2} + 8 a + 3\right)\cdot 19^{5} + \left(3 a^{3} + 5 a^{2} + 17 a + 6\right)\cdot 19^{6} + \left(18 a^{3} + 12 a^{2} + 14 a + 17\right)\cdot 19^{7} + \left(2 a^{3} + 9 a^{2} + 3 a + 16\right)\cdot 19^{8} + \left(6 a^{3} + 8 a^{2} + 5 a + 16\right)\cdot 19^{9} +O(19^{10})$$ 2*a^3 + 13*a^2 + 15*a + 16 + (2*a^3 + 11*a^2 + 17*a + 17)*19 + (8*a^3 + 13*a^2 + 15*a + 12)*19^2 + (17*a^3 + 16*a^2 + 10*a)*19^3 + (12*a^3 + a^2 + 3*a + 17)*19^4 + (13*a^3 + 15*a^2 + 8*a + 3)*19^5 + (3*a^3 + 5*a^2 + 17*a + 6)*19^6 + (18*a^3 + 12*a^2 + 14*a + 17)*19^7 + (2*a^3 + 9*a^2 + 3*a + 16)*19^8 + (6*a^3 + 8*a^2 + 5*a + 16)*19^9+O(19^10) $r_{ 7 }$ $=$ $$8 a^{3} + 3 a^{2} + 17 a + 7 + \left(7 a^{3} + 2 a^{2} + 9 a + 16\right)\cdot 19 + \left(4 a^{3} + 2 a^{2} + 4 a + 13\right)\cdot 19^{2} + \left(7 a^{3} + 7 a^{2} + 10 a + 12\right)\cdot 19^{3} + \left(7 a^{3} + 8 a^{2} + 2 a + 18\right)\cdot 19^{4} + \left(4 a^{2} + 3 a + 13\right)\cdot 19^{5} + \left(3 a^{3} + 6 a^{2} + 10 a + 5\right)\cdot 19^{6} + \left(11 a^{3} + a^{2} + 15 a + 13\right)\cdot 19^{7} + \left(16 a^{3} + 5 a^{2} + 11 a + 13\right)\cdot 19^{8} + \left(18 a^{3} + 11 a^{2} + 4 a + 10\right)\cdot 19^{9} +O(19^{10})$$ 8*a^3 + 3*a^2 + 17*a + 7 + (7*a^3 + 2*a^2 + 9*a + 16)*19 + (4*a^3 + 2*a^2 + 4*a + 13)*19^2 + (7*a^3 + 7*a^2 + 10*a + 12)*19^3 + (7*a^3 + 8*a^2 + 2*a + 18)*19^4 + (4*a^2 + 3*a + 13)*19^5 + (3*a^3 + 6*a^2 + 10*a + 5)*19^6 + (11*a^3 + a^2 + 15*a + 13)*19^7 + (16*a^3 + 5*a^2 + 11*a + 13)*19^8 + (18*a^3 + 11*a^2 + 4*a + 10)*19^9+O(19^10) $r_{ 8 }$ $=$ $$6 a^{3} + 16 a + 17 + \left(17 a^{3} + 9 a^{2} + 10 a + 2\right)\cdot 19 + \left(2 a^{3} + 15 a^{2} + 9 a + 5\right)\cdot 19^{2} + \left(5 a^{3} + 17 a^{2} + 17 a\right)\cdot 19^{3} + \left(14 a^{3} + 9 a^{2} + 7 a + 3\right)\cdot 19^{4} + \left(3 a^{3} + a^{2} + 6 a + 8\right)\cdot 19^{5} + \left(11 a^{3} + 3 a^{2} + 12 a + 3\right)\cdot 19^{6} + \left(13 a^{3} + 11 a^{2} + 18 a + 16\right)\cdot 19^{7} + \left(4 a^{3} + 11 a^{2} + 5 a + 4\right)\cdot 19^{8} + \left(13 a^{2} + 15 a + 11\right)\cdot 19^{9} +O(19^{10})$$ 6*a^3 + 16*a + 17 + (17*a^3 + 9*a^2 + 10*a + 2)*19 + (2*a^3 + 15*a^2 + 9*a + 5)*19^2 + (5*a^3 + 17*a^2 + 17*a)*19^3 + (14*a^3 + 9*a^2 + 7*a + 3)*19^4 + (3*a^3 + a^2 + 6*a + 8)*19^5 + (11*a^3 + 3*a^2 + 12*a + 3)*19^6 + (13*a^3 + 11*a^2 + 18*a + 16)*19^7 + (4*a^3 + 11*a^2 + 5*a + 4)*19^8 + (13*a^2 + 15*a + 11)*19^9+O(19^10) $r_{ 9 }$ $=$ $$9 a^{3} + 4 a^{2} + 6 a + 3 + \left(12 a^{3} + 9 a^{2} + 14 a + 1\right)\cdot 19 + \left(10 a^{3} + 2 a^{2} + 9 a + 18\right)\cdot 19^{2} + \left(4 a^{3} + 10 a^{2} + 16 a + 1\right)\cdot 19^{3} + \left(14 a^{3} + 7 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(15 a^{3} + 4 a^{2} + 14 a + 10\right)\cdot 19^{5} + \left(15 a^{3} + 9 a^{2} + 18 a + 14\right)\cdot 19^{6} + \left(7 a^{3} + 9 a^{2} + 18 a + 9\right)\cdot 19^{7} + \left(3 a^{3} + 16 a^{2} + 3 a + 8\right)\cdot 19^{8} + \left(a^{3} + 13 a + 1\right)\cdot 19^{9} +O(19^{10})$$ 9*a^3 + 4*a^2 + 6*a + 3 + (12*a^3 + 9*a^2 + 14*a + 1)*19 + (10*a^3 + 2*a^2 + 9*a + 18)*19^2 + (4*a^3 + 10*a^2 + 16*a + 1)*19^3 + (14*a^3 + 7*a^2 + 2*a + 5)*19^4 + (15*a^3 + 4*a^2 + 14*a + 10)*19^5 + (15*a^3 + 9*a^2 + 18*a + 14)*19^6 + (7*a^3 + 9*a^2 + 18*a + 9)*19^7 + (3*a^3 + 16*a^2 + 3*a + 8)*19^8 + (a^3 + 13*a + 1)*19^9+O(19^10)

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

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

### Character values on conjugacy classes

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