# Properties

 Label 8.640...303.24t1539.a Dimension $8$ Group $S_3 \wr C_3$ Conductor $6.405\times 10^{14}$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $8$ Group: $S_3 \wr C_3$ Conductor: $$640548715585303$$$$\medspace = 7^{7} \cdot 167^{4}$$ Artin number field: Galois closure of 9.3.962721767.1 Galois orbit size: $2$ Smallest permutation container: 24T1539 Parity: even Projective image: $S_3\wr C_3$ Projective field: Galois closure of 9.3.962721767.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $8$ $8$ $9$ $2$ $(1,2)$ $0$ $0$ $27$ $2$ $(1,2)(3,4)(5,8)$ $0$ $0$ $27$ $2$ $(1,2)(8,9)$ $0$ $0$ $6$ $3$ $(3,4,6)$ $-4$ $-4$ $8$ $3$ $(1,2,7)(3,4,6)(5,8,9)$ $-1$ $-1$ $12$ $3$ $(3,4,6)(5,8,9)$ $2$ $2$ $36$ $3$ $(1,5,3)(2,8,4)(6,7,9)$ $2 \zeta_{3}$ $-2 \zeta_{3} - 2$ $36$ $3$ $(1,3,5)(2,4,8)(6,9,7)$ $-2 \zeta_{3} - 2$ $2 \zeta_{3}$ $18$ $6$ $(1,2)(3,4,6)$ $0$ $0$ $18$ $6$ $(1,2)(5,8,9)$ $0$ $0$ $36$ $6$ $(1,2)(3,4,6)(5,8,9)$ $0$ $0$ $54$ $6$ $(1,2)(3,4,6)(8,9)$ $0$ $0$ $108$ $6$ $(1,8,4,2,5,3)(6,7,9)$ $0$ $0$ $108$ $6$ $(1,3,5,2,4,8)(6,9,7)$ $0$ $0$ $72$ $9$ $(1,5,3,2,8,4,7,9,6)$ $-\zeta_{3}$ $\zeta_{3} + 1$ $72$ $9$ $(1,3,8,7,6,5,2,4,9)$ $\zeta_{3} + 1$ $-\zeta_{3}$
The blue line marks the conjugacy class containing complex conjugation.