# Properties

 Label 12.171...241.18t206.a Dimension $12$ Group $S_3 \wr C_3$ Conductor $1.711\times 10^{17}$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3 \wr C_3$ Conductor: $$171126940264356241$$$$\medspace = 11^{4} \cdot 43^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.3.69534993539.1 Galois orbit size: $1$ Smallest permutation container: 18T206 Parity: even Projective image: $S_3\wr C_3$ Projective field: Galois closure of 9.3.69534993539.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{3} + 6x + 35$$
Roots:
 $r_{ 1 }$ $=$ $$3 a^{2} + 15 a + 28 + \left(9 a^{2} + 9 a + 3\right)\cdot 37 + \left(12 a^{2} + 22 a + 34\right)\cdot 37^{2} + \left(26 a^{2} + 20 a + 31\right)\cdot 37^{3} + \left(22 a^{2} + 3 a + 9\right)\cdot 37^{4} + \left(34 a^{2} + 29 a + 4\right)\cdot 37^{5} + \left(14 a^{2} + 31 a\right)\cdot 37^{6} + \left(23 a^{2} + 22 a + 9\right)\cdot 37^{7} + \left(28 a^{2} + 29 a + 9\right)\cdot 37^{8} + \left(10 a^{2} + 11 a + 12\right)\cdot 37^{9} +O(37^{10})$$ 3*a^2 + 15*a + 28 + (9*a^2 + 9*a + 3)*37 + (12*a^2 + 22*a + 34)*37^2 + (26*a^2 + 20*a + 31)*37^3 + (22*a^2 + 3*a + 9)*37^4 + (34*a^2 + 29*a + 4)*37^5 + (14*a^2 + 31*a)*37^6 + (23*a^2 + 22*a + 9)*37^7 + (28*a^2 + 29*a + 9)*37^8 + (10*a^2 + 11*a + 12)*37^9+O(37^10) $r_{ 2 }$ $=$ $$12 a^{2} + 13 a + 34 + \left(21 a^{2} + 5 a + 21\right)\cdot 37 + \left(5 a^{2} + 9 a + 29\right)\cdot 37^{2} + \left(36 a^{2} + 13 a + 28\right)\cdot 37^{3} + \left(22 a^{2} + 18 a + 36\right)\cdot 37^{4} + \left(24 a^{2} + 35 a + 16\right)\cdot 37^{5} + \left(23 a^{2} + 16 a + 34\right)\cdot 37^{6} + \left(12 a^{2} + 7 a + 26\right)\cdot 37^{7} + \left(30 a^{2} + 27 a + 10\right)\cdot 37^{8} + \left(35 a^{2} + 4\right)\cdot 37^{9} +O(37^{10})$$ 12*a^2 + 13*a + 34 + (21*a^2 + 5*a + 21)*37 + (5*a^2 + 9*a + 29)*37^2 + (36*a^2 + 13*a + 28)*37^3 + (22*a^2 + 18*a + 36)*37^4 + (24*a^2 + 35*a + 16)*37^5 + (23*a^2 + 16*a + 34)*37^6 + (12*a^2 + 7*a + 26)*37^7 + (30*a^2 + 27*a + 10)*37^8 + (35*a^2 + 4)*37^9+O(37^10) $r_{ 3 }$ $=$ $$19 a^{2} + 13 a + 18 + \left(25 a^{2} + 27 a + 32\right)\cdot 37 + \left(24 a^{2} + 12 a + 9\right)\cdot 37^{2} + \left(5 a^{2} + 35 a + 23\right)\cdot 37^{3} + \left(6 a^{2} + 4 a + 17\right)\cdot 37^{4} + \left(a^{2} + 27 a + 18\right)\cdot 37^{5} + \left(12 a^{2} + 14 a + 25\right)\cdot 37^{6} + \left(24 a^{2} + 10 a + 12\right)\cdot 37^{7} + \left(21 a^{2} + 17 a + 18\right)\cdot 37^{8} + \left(14 a^{2} + 13 a + 27\right)\cdot 37^{9} +O(37^{10})$$ 19*a^2 + 13*a + 18 + (25*a^2 + 27*a + 32)*37 + (24*a^2 + 12*a + 9)*37^2 + (5*a^2 + 35*a + 23)*37^3 + (6*a^2 + 4*a + 17)*37^4 + (a^2 + 27*a + 18)*37^5 + (12*a^2 + 14*a + 25)*37^6 + (24*a^2 + 10*a + 12)*37^7 + (21*a^2 + 17*a + 18)*37^8 + (14*a^2 + 13*a + 27)*37^9+O(37^10) $r_{ 4 }$ $=$ $$19 a^{2} + 2 a + 25 + \left(5 a^{2} + 8 a + 32\right)\cdot 37 + \left(13 a^{2} + 33 a + 22\right)\cdot 37^{2} + \left(29 a^{2} + 8 a + 1\right)\cdot 37^{3} + \left(32 a^{2} + 17 a + 2\right)\cdot 37^{4} + \left(5 a^{2} + 17 a + 16\right)\cdot 37^{5} + \left(9 a^{2} + 17 a + 13\right)\cdot 37^{6} + \left(21 a^{2} + 14 a + 24\right)\cdot 37^{7} + \left(26 a^{2} + 28 a + 32\right)\cdot 37^{8} + \left(14 a^{2} + 3 a + 30\right)\cdot 37^{9} +O(37^{10})$$ 19*a^2 + 2*a + 25 + (5*a^2 + 8*a + 32)*37 + (13*a^2 + 33*a + 22)*37^2 + (29*a^2 + 8*a + 1)*37^3 + (32*a^2 + 17*a + 2)*37^4 + (5*a^2 + 17*a + 16)*37^5 + (9*a^2 + 17*a + 13)*37^6 + (21*a^2 + 14*a + 24)*37^7 + (26*a^2 + 28*a + 32)*37^8 + (14*a^2 + 3*a + 30)*37^9+O(37^10) $r_{ 5 }$ $=$ $$10 a^{2} + 15 a + 1 + \left(13 a^{2} + 31 a + 1\right)\cdot 37 + \left(31 a^{2} + 25 a + 22\right)\cdot 37^{2} + \left(32 a^{2} + 5 a + 24\right)\cdot 37^{3} + \left(5 a^{2} + 27 a + 11\right)\cdot 37^{4} + \left(11 a^{2} + 20 a + 1\right)\cdot 37^{5} + \left(3 a^{2} + 29 a + 22\right)\cdot 37^{6} + \left(35 a^{2} + 25 a + 26\right)\cdot 37^{7} + \left(19 a^{2} + 19 a + 36\right)\cdot 37^{8} + \left(26 a^{2} + 24 a + 16\right)\cdot 37^{9} +O(37^{10})$$ 10*a^2 + 15*a + 1 + (13*a^2 + 31*a + 1)*37 + (31*a^2 + 25*a + 22)*37^2 + (32*a^2 + 5*a + 24)*37^3 + (5*a^2 + 27*a + 11)*37^4 + (11*a^2 + 20*a + 1)*37^5 + (3*a^2 + 29*a + 22)*37^6 + (35*a^2 + 25*a + 26)*37^7 + (19*a^2 + 19*a + 36)*37^8 + (26*a^2 + 24*a + 16)*37^9+O(37^10) $r_{ 6 }$ $=$ $$6 a^{2} + 22 a + 10 + \left(10 a^{2} + 23 a + 14\right)\cdot 37 + \left(18 a^{2} + 31 a + 6\right)\cdot 37^{2} + \left(8 a^{2} + 14 a + 29\right)\cdot 37^{3} + \left(18 a^{2} + a + 17\right)\cdot 37^{4} + \left(6 a^{2} + 21 a + 18\right)\cdot 37^{5} + \left(4 a^{2} + 2 a + 30\right)\cdot 37^{6} + \left(3 a^{2} + 15 a + 25\right)\cdot 37^{7} + \left(17 a^{2} + 18 a + 31\right)\cdot 37^{8} + \left(23 a^{2} + 32 a + 28\right)\cdot 37^{9} +O(37^{10})$$ 6*a^2 + 22*a + 10 + (10*a^2 + 23*a + 14)*37 + (18*a^2 + 31*a + 6)*37^2 + (8*a^2 + 14*a + 29)*37^3 + (18*a^2 + a + 17)*37^4 + (6*a^2 + 21*a + 18)*37^5 + (4*a^2 + 2*a + 30)*37^6 + (3*a^2 + 15*a + 25)*37^7 + (17*a^2 + 18*a + 31)*37^8 + (23*a^2 + 32*a + 28)*37^9+O(37^10) $r_{ 7 }$ $=$ $$15 a^{2} + 9 a + 2 + \left(2 a^{2} + 14\right)\cdot 37 + \left(2 a + 22\right)\cdot 37^{2} + \left(5 a^{2} + 18 a + 20\right)\cdot 37^{3} + \left(8 a^{2} + 28 a + 25\right)\cdot 37^{4} + \left(a^{2} + 17 a + 18\right)\cdot 37^{5} + \left(10 a^{2} + 27 a + 17\right)\cdot 37^{6} + \left(26 a^{2} + 3 a + 20\right)\cdot 37^{7} + \left(23 a^{2} + 27 a + 26\right)\cdot 37^{8} + \left(11 a^{2} + 11 a + 15\right)\cdot 37^{9} +O(37^{10})$$ 15*a^2 + 9*a + 2 + (2*a^2 + 14)*37 + (2*a + 22)*37^2 + (5*a^2 + 18*a + 20)*37^3 + (8*a^2 + 28*a + 25)*37^4 + (a^2 + 17*a + 18)*37^5 + (10*a^2 + 27*a + 17)*37^6 + (26*a^2 + 3*a + 20)*37^7 + (23*a^2 + 27*a + 26)*37^8 + (11*a^2 + 11*a + 15)*37^9+O(37^10) $r_{ 8 }$ $=$ $$28 a^{2} + 36 + \left(17 a^{2} + 4 a + 18\right)\cdot 37 + \left(6 a^{2} + 20 a + 33\right)\cdot 37^{2} + \left(2 a^{2} + a + 12\right)\cdot 37^{3} + \left(33 a^{2} + 32 a + 9\right)\cdot 37^{4} + \left(32 a^{2} + 23 a + 14\right)\cdot 37^{5} + \left(17 a^{2} + 2 a + 6\right)\cdot 37^{6} + \left(10 a^{2} + 36 a + 2\right)\cdot 37^{7} + \left(28 a^{2} + 25 a + 33\right)\cdot 37^{8} + \left(2 a^{2} + 29 a + 32\right)\cdot 37^{9} +O(37^{10})$$ 28*a^2 + 36 + (17*a^2 + 4*a + 18)*37 + (6*a^2 + 20*a + 33)*37^2 + (2*a^2 + a + 12)*37^3 + (33*a^2 + 32*a + 9)*37^4 + (32*a^2 + 23*a + 14)*37^5 + (17*a^2 + 2*a + 6)*37^6 + (10*a^2 + 36*a + 2)*37^7 + (28*a^2 + 25*a + 33)*37^8 + (2*a^2 + 29*a + 32)*37^9+O(37^10) $r_{ 9 }$ $=$ $$36 a^{2} + 22 a + 31 + \left(5 a^{2} + a + 8\right)\cdot 37 + \left(36 a^{2} + 28 a + 4\right)\cdot 37^{2} + \left(a^{2} + 29 a + 12\right)\cdot 37^{3} + \left(35 a^{2} + 14 a + 17\right)\cdot 37^{4} + \left(29 a^{2} + 29 a + 2\right)\cdot 37^{5} + \left(15 a^{2} + 4 a + 35\right)\cdot 37^{6} + \left(28 a^{2} + 12 a + 36\right)\cdot 37^{7} + \left(25 a^{2} + 28 a + 22\right)\cdot 37^{8} + \left(7 a^{2} + 19 a + 15\right)\cdot 37^{9} +O(37^{10})$$ 36*a^2 + 22*a + 31 + (5*a^2 + a + 8)*37 + (36*a^2 + 28*a + 4)*37^2 + (a^2 + 29*a + 12)*37^3 + (35*a^2 + 14*a + 17)*37^4 + (29*a^2 + 29*a + 2)*37^5 + (15*a^2 + 4*a + 35)*37^6 + (28*a^2 + 12*a + 36)*37^7 + (25*a^2 + 28*a + 22)*37^8 + (7*a^2 + 19*a + 15)*37^9+O(37^10)

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

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

### Character values on conjugacy classes

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