# Properties

 Label 12.373...032.36t2216.a Dimension $12$ Group $S_3\wr S_3$ Conductor $3.737\times 10^{17}$ Indicator $1$

# Learn more

## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$373677297806380032$$$$\medspace = 2^{10} \cdot 3^{17} \cdot 41^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 9.1.5578004736.1 Galois orbit size: $1$ Smallest permutation container: 36T2216 Parity: odd Projective image: $S_3\wr S_3$ Projective field: Galois closure of 9.1.5578004736.1

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character values $c1$ $1$ $1$ $()$ $12$ $9$ $2$ $(3,6)$ $4$ $18$ $2$ $(2,3)(5,6)(8,9)$ $-2$ $27$ $2$ $(1,4)(2,5)(3,6)$ $0$ $27$ $2$ $(1,4)(3,6)$ $0$ $54$ $2$ $(1,2)(3,6)(4,5)(7,8)$ $-2$ $6$ $3$ $(1,4,7)$ $0$ $8$ $3$ $(1,7,4)(2,8,5)(3,9,6)$ $3$ $12$ $3$ $(1,4,7)(2,5,8)$ $-3$ $72$ $3$ $(1,3,2)(4,6,5)(7,9,8)$ $0$ $54$ $4$ $(1,6,4,3)(7,9)$ $0$ $162$ $4$ $(1,6,4,3)(5,8)(7,9)$ $0$ $36$ $6$ $(1,4,7)(2,3)(5,6)(8,9)$ $-2$ $36$ $6$ $(1,9,7,6,4,3)$ $1$ $36$ $6$ $(1,4,7)(3,6)$ $-2$ $36$ $6$ $(1,4,7)(2,5,8)(3,6)$ $1$ $54$ $6$ $(1,4)(2,8,5)(3,6)$ $0$ $72$ $6$ $(1,4,7)(2,6,5,9,8,3)$ $1$ $108$ $6$ $(1,5,4,8,7,2)(3,6)$ $1$ $216$ $6$ $(1,6,5,4,3,2)(7,9,8)$ $0$ $144$ $9$ $(1,9,8,7,6,5,4,3,2)$ $0$ $108$ $12$ $(1,6,4,3)(2,5,8)(7,9)$ $0$
The blue line marks the conjugacy class containing complex conjugation.