# Properties

 Label 6.2472128.9t31.a.a Dimension $6$ Group $S_3\wr S_3$ Conductor $2472128$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $6$ Group: $S_3\wr S_3$ Conductor: $$2472128$$$$\medspace = 2^{6} \cdot 19^{2} \cdot 107$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.1.187881728.1 Galois orbit size: $1$ Smallest permutation container: $S_3\wr S_3$ Parity: odd Determinant: 1.107.2t1.a.a Projective image: $S_3\wr S_3$ Projective stem field: Galois closure of 9.1.187881728.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - x^{8} - 2x^{7} + 2x^{5} + 2x^{3} + x + 1$$ x^9 - x^8 - 2*x^7 + 2*x^5 + 2*x^3 + x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $$x^{3} + 7x + 59$$

Roots:
 $r_{ 1 }$ $=$ $$49 a^{2} + 31 a + 9 + \left(49 a^{2} + 22 a + 5\right)\cdot 61 + \left(41 a^{2} + 29 a + 5\right)\cdot 61^{2} + \left(46 a^{2} + 22 a + 31\right)\cdot 61^{3} + \left(17 a^{2} + 20 a + 28\right)\cdot 61^{4} + \left(36 a^{2} + 33 a + 58\right)\cdot 61^{5} + \left(22 a^{2} + 4 a + 28\right)\cdot 61^{6} + \left(25 a^{2} + 11 a + 37\right)\cdot 61^{7} + \left(57 a^{2} + 29 a + 58\right)\cdot 61^{8} + \left(60 a + 5\right)\cdot 61^{9} +O(61^{10})$$ 49*a^2 + 31*a + 9 + (49*a^2 + 22*a + 5)*61 + (41*a^2 + 29*a + 5)*61^2 + (46*a^2 + 22*a + 31)*61^3 + (17*a^2 + 20*a + 28)*61^4 + (36*a^2 + 33*a + 58)*61^5 + (22*a^2 + 4*a + 28)*61^6 + (25*a^2 + 11*a + 37)*61^7 + (57*a^2 + 29*a + 58)*61^8 + (60*a + 5)*61^9+O(61^10) $r_{ 2 }$ $=$ $$7 a^{2} + 10 a + 28 + \left(20 a^{2} + 18 a + 21\right)\cdot 61 + \left(46 a^{2} + 2 a + 41\right)\cdot 61^{2} + \left(13 a^{2} + 49 a + 26\right)\cdot 61^{3} + \left(21 a^{2} + 36 a + 60\right)\cdot 61^{4} + \left(52 a^{2} + 49 a + 31\right)\cdot 61^{5} + \left(44 a^{2} + 8 a + 42\right)\cdot 61^{6} + \left(25 a^{2} + 6 a + 29\right)\cdot 61^{7} + \left(44 a^{2} + 24 a + 28\right)\cdot 61^{8} + \left(58 a^{2} + 6 a + 19\right)\cdot 61^{9} +O(61^{10})$$ 7*a^2 + 10*a + 28 + (20*a^2 + 18*a + 21)*61 + (46*a^2 + 2*a + 41)*61^2 + (13*a^2 + 49*a + 26)*61^3 + (21*a^2 + 36*a + 60)*61^4 + (52*a^2 + 49*a + 31)*61^5 + (44*a^2 + 8*a + 42)*61^6 + (25*a^2 + 6*a + 29)*61^7 + (44*a^2 + 24*a + 28)*61^8 + (58*a^2 + 6*a + 19)*61^9+O(61^10) $r_{ 3 }$ $=$ $$12 a^{2} + 31 a + 57 + \left(24 a^{2} + 34 a + 5\right)\cdot 61 + \left(43 a^{2} + 7 a + 59\right)\cdot 61^{2} + \left(7 a^{2} + 29 a + 56\right)\cdot 61^{3} + \left(34 a^{2} + 3 a + 48\right)\cdot 61^{4} + \left(29 a^{2} + 27 a + 13\right)\cdot 61^{5} + \left(16 a^{2} + 48 a + 15\right)\cdot 61^{6} + \left(29 a^{2} + 51 a + 23\right)\cdot 61^{7} + \left(55 a^{2} + 27 a + 57\right)\cdot 61^{8} + \left(a^{2} + 19 a + 17\right)\cdot 61^{9} +O(61^{10})$$ 12*a^2 + 31*a + 57 + (24*a^2 + 34*a + 5)*61 + (43*a^2 + 7*a + 59)*61^2 + (7*a^2 + 29*a + 56)*61^3 + (34*a^2 + 3*a + 48)*61^4 + (29*a^2 + 27*a + 13)*61^5 + (16*a^2 + 48*a + 15)*61^6 + (29*a^2 + 51*a + 23)*61^7 + (55*a^2 + 27*a + 57)*61^8 + (a^2 + 19*a + 17)*61^9+O(61^10) $r_{ 4 }$ $=$ $$8 a^{2} + 26 a + 21 + \left(33 a^{2} + 52 a + 49\right)\cdot 61 + \left(40 a^{2} + 53 a + 39\right)\cdot 61^{2} + \left(6 a^{2} + 47 a + 47\right)\cdot 61^{3} + \left(35 a^{2} + 30 a + 7\right)\cdot 61^{4} + \left(12 a^{2} + 27 a + 29\right)\cdot 61^{5} + \left(30 a^{2} + 55 a + 3\right)\cdot 61^{6} + \left(47 a^{2} + 6 a + 39\right)\cdot 61^{7} + \left(60 a^{2} + 3 a + 33\right)\cdot 61^{8} + \left(52 a^{2} + 49 a + 45\right)\cdot 61^{9} +O(61^{10})$$ 8*a^2 + 26*a + 21 + (33*a^2 + 52*a + 49)*61 + (40*a^2 + 53*a + 39)*61^2 + (6*a^2 + 47*a + 47)*61^3 + (35*a^2 + 30*a + 7)*61^4 + (12*a^2 + 27*a + 29)*61^5 + (30*a^2 + 55*a + 3)*61^6 + (47*a^2 + 6*a + 39)*61^7 + (60*a^2 + 3*a + 33)*61^8 + (52*a^2 + 49*a + 45)*61^9+O(61^10) $r_{ 5 }$ $=$ $$8 a^{2} + 5 a + 18 + \left(56 a^{2} + 25 a + 53\right)\cdot 61 + \left(9 a^{2} + 27 a + 24\right)\cdot 61^{2} + \left(36 a^{2} + 48 a + 47\right)\cdot 61^{3} + \left(22 a^{2} + a + 35\right)\cdot 61^{4} + \left(11 a^{2} + 36 a + 30\right)\cdot 61^{5} + \left(34 a^{2} + 6 a + 16\right)\cdot 61^{6} + \left(44 a^{2} + 57 a + 13\right)\cdot 61^{7} + \left(23 a^{2} + 33 a + 31\right)\cdot 61^{8} + \left(38 a^{2} + 25 a + 25\right)\cdot 61^{9} +O(61^{10})$$ 8*a^2 + 5*a + 18 + (56*a^2 + 25*a + 53)*61 + (9*a^2 + 27*a + 24)*61^2 + (36*a^2 + 48*a + 47)*61^3 + (22*a^2 + a + 35)*61^4 + (11*a^2 + 36*a + 30)*61^5 + (34*a^2 + 6*a + 16)*61^6 + (44*a^2 + 57*a + 13)*61^7 + (23*a^2 + 33*a + 31)*61^8 + (38*a^2 + 25*a + 25)*61^9+O(61^10) $r_{ 6 }$ $=$ $$50 a^{2} + 42 a + 5 + \left(24 a^{2} + 2 a + 23\right)\cdot 61 + \left(5 a^{2} + 47 a + 33\right)\cdot 61^{2} + \left(4 a^{2} + 34 a + 42\right)\cdot 61^{3} + \left(12 a^{2} + 60 a + 17\right)\cdot 61^{4} + \left(25 a^{2} + 43 a + 27\right)\cdot 61^{5} + \left(19 a + 38\right)\cdot 61^{6} + \left(54 a^{2} + 17 a + 39\right)\cdot 61^{7} + \left(13 a^{2} + 49 a + 48\right)\cdot 61^{8} + \left(8 a^{2} + 45 a + 27\right)\cdot 61^{9} +O(61^{10})$$ 50*a^2 + 42*a + 5 + (24*a^2 + 2*a + 23)*61 + (5*a^2 + 47*a + 33)*61^2 + (4*a^2 + 34*a + 42)*61^3 + (12*a^2 + 60*a + 17)*61^4 + (25*a^2 + 43*a + 27)*61^5 + (19*a + 38)*61^6 + (54*a^2 + 17*a + 39)*61^7 + (13*a^2 + 49*a + 48)*61^8 + (8*a^2 + 45*a + 27)*61^9+O(61^10) $r_{ 7 }$ $=$ $$41 a^{2} + 25 a + 50 + \left(41 a^{2} + a + 46\right)\cdot 61 + \left(7 a^{2} + 26 a + 34\right)\cdot 61^{2} + \left(17 a^{2} + 44 a + 19\right)\cdot 61^{3} + \left(4 a^{2} + 55 a + 11\right)\cdot 61^{4} + \left(20 a^{2} + 58 a + 10\right)\cdot 61^{5} + \left(10 a^{2} + 5 a + 27\right)\cdot 61^{6} + \left(48 a^{2} + 13 a + 50\right)\cdot 61^{7} + \left(42 a^{2} + 60 a + 38\right)\cdot 61^{8} + \left(20 a^{2} + 15 a + 44\right)\cdot 61^{9} +O(61^{10})$$ 41*a^2 + 25*a + 50 + (41*a^2 + a + 46)*61 + (7*a^2 + 26*a + 34)*61^2 + (17*a^2 + 44*a + 19)*61^3 + (4*a^2 + 55*a + 11)*61^4 + (20*a^2 + 58*a + 10)*61^5 + (10*a^2 + 5*a + 27)*61^6 + (48*a^2 + 13*a + 50)*61^7 + (42*a^2 + 60*a + 38)*61^8 + (20*a^2 + 15*a + 44)*61^9+O(61^10) $r_{ 8 }$ $=$ $$4 a^{2} + 4 a + 43 + \left(39 a^{2} + 47 a + 56\right)\cdot 61 + \left(39 a^{2} + 38 a + 55\right)\cdot 61^{2} + \left(7 a^{2} + 51 a + 31\right)\cdot 61^{3} + \left(8 a^{2} + 9 a + 44\right)\cdot 61^{4} + \left(12 a^{2} + 6\right)\cdot 61^{5} + \left(8 a^{2} + a + 43\right)\cdot 61^{6} + \left(49 a^{2} + 43 a + 46\right)\cdot 61^{7} + \left(3 a^{2} + 28 a + 11\right)\cdot 61^{8} + \left(7 a^{2} + 12 a + 14\right)\cdot 61^{9} +O(61^{10})$$ 4*a^2 + 4*a + 43 + (39*a^2 + 47*a + 56)*61 + (39*a^2 + 38*a + 55)*61^2 + (7*a^2 + 51*a + 31)*61^3 + (8*a^2 + 9*a + 44)*61^4 + (12*a^2 + 6)*61^5 + (8*a^2 + a + 43)*61^6 + (49*a^2 + 43*a + 46)*61^7 + (3*a^2 + 28*a + 11)*61^8 + (7*a^2 + 12*a + 14)*61^9+O(61^10) $r_{ 9 }$ $=$ $$4 a^{2} + 9 a + 14 + \left(16 a^{2} + 40 a + 43\right)\cdot 61 + \left(9 a^{2} + 11 a + 10\right)\cdot 61^{2} + \left(43 a^{2} + 38 a + 1\right)\cdot 61^{3} + \left(27 a^{2} + 24 a + 50\right)\cdot 61^{4} + \left(44 a^{2} + 28 a + 35\right)\cdot 61^{5} + \left(15 a^{2} + 32 a + 28\right)\cdot 61^{6} + \left(42 a^{2} + 37 a + 25\right)\cdot 61^{7} + \left(2 a^{2} + 48 a + 57\right)\cdot 61^{8} + \left(55 a^{2} + 8 a + 42\right)\cdot 61^{9} +O(61^{10})$$ 4*a^2 + 9*a + 14 + (16*a^2 + 40*a + 43)*61 + (9*a^2 + 11*a + 10)*61^2 + (43*a^2 + 38*a + 1)*61^3 + (27*a^2 + 24*a + 50)*61^4 + (44*a^2 + 28*a + 35)*61^5 + (15*a^2 + 32*a + 28)*61^6 + (42*a^2 + 37*a + 25)*61^7 + (2*a^2 + 48*a + 57)*61^8 + (55*a^2 + 8*a + 42)*61^9+O(61^10)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $6$ $9$ $2$ $(1,5)$ $4$ $18$ $2$ $(1,2)(5,7)(8,9)$ $2$ $27$ $2$ $(1,5)(2,7)(3,4)$ $0$ $27$ $2$ $(1,5)(2,7)$ $2$ $54$ $2$ $(1,5)(2,3)(4,7)(6,8)$ $0$ $6$ $3$ $(3,4,6)$ $3$ $8$ $3$ $(1,5,9)(2,7,8)(3,4,6)$ $-3$ $12$ $3$ $(2,7,8)(3,4,6)$ $0$ $72$ $3$ $(1,2,3)(4,5,7)(6,9,8)$ $0$ $54$ $4$ $(1,7,5,2)(8,9)$ $2$ $162$ $4$ $(1,4,5,3)(6,9)(7,8)$ $0$ $36$ $6$ $(1,2)(3,4,6)(5,7)(8,9)$ $-1$ $36$ $6$ $(1,3,5,4,9,6)$ $2$ $36$ $6$ $(1,5)(3,4,6)$ $1$ $36$ $6$ $(1,5)(2,7,8)(3,4,6)$ $-2$ $54$ $6$ $(1,5)(2,7)(3,6,4)$ $-1$ $72$ $6$ $(1,2,5,7,9,8)(3,4,6)$ $-1$ $108$ $6$ $(1,5)(2,3,7,4,8,6)$ $0$ $216$ $6$ $(1,7,4,5,2,3)(6,9,8)$ $0$ $144$ $9$ $(1,2,3,5,7,4,9,8,6)$ $0$ $108$ $12$ $(1,7,5,2)(3,4,6)(8,9)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.