# Properties

 Label 12.198...088.18t315.a.a Dimension $12$ Group $S_3\wr S_3$ Conductor $1.987\times 10^{16}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3\wr S_3$ Conductor: $$19871009045337088$$$$\medspace = 2^{10} \cdot 23^{4} \cdot 37^{5}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.11035059968.1 Galois orbit size: $1$ Smallest permutation container: 18T315 Parity: even Determinant: 1.37.2t1.a.a Projective image: $C_3^3.S_4.C_2$ Projective stem field: Galois closure of 9.3.11035059968.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{8} + 2x^{7} - 2x^{6} + 6x^{5} - 2x^{4} + 2x^{3} - 6x^{2} - 3x + 1$$ x^9 - 3*x^8 + 2*x^7 - 2*x^6 + 6*x^5 - 2*x^4 + 2*x^3 - 6*x^2 - 3*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $$x^{3} + 2x + 68$$

Roots:
 $r_{ 1 }$ $=$ $$36 a^{2} + 16 a + 40 + \left(58 a^{2} + 39 a + 33\right)\cdot 73 + \left(7 a^{2} + 20 a + 13\right)\cdot 73^{2} + \left(6 a^{2} + 39 a + 9\right)\cdot 73^{3} + \left(44 a^{2} + 21 a + 4\right)\cdot 73^{4} + \left(70 a^{2} + 31 a + 58\right)\cdot 73^{5} + \left(40 a^{2} + 34 a + 11\right)\cdot 73^{6} + \left(54 a^{2} + 24 a + 24\right)\cdot 73^{7} + \left(7 a^{2} + 38 a + 44\right)\cdot 73^{8} + \left(30 a^{2} + 58 a + 17\right)\cdot 73^{9} +O(73^{10})$$ 36*a^2 + 16*a + 40 + (58*a^2 + 39*a + 33)*73 + (7*a^2 + 20*a + 13)*73^2 + (6*a^2 + 39*a + 9)*73^3 + (44*a^2 + 21*a + 4)*73^4 + (70*a^2 + 31*a + 58)*73^5 + (40*a^2 + 34*a + 11)*73^6 + (54*a^2 + 24*a + 24)*73^7 + (7*a^2 + 38*a + 44)*73^8 + (30*a^2 + 58*a + 17)*73^9+O(73^10) $r_{ 2 }$ $=$ $$65 a^{2} + 15 a + 30 + \left(45 a^{2} + 46 a + 65\right)\cdot 73 + \left(22 a^{2} + 58 a + 8\right)\cdot 73^{2} + \left(8 a^{2} + 22 a + 12\right)\cdot 73^{3} + \left(51 a^{2} + 71 a + 62\right)\cdot 73^{4} + \left(39 a^{2} + 40 a + 16\right)\cdot 73^{5} + \left(35 a^{2} + 67 a + 53\right)\cdot 73^{6} + \left(42 a^{2} + 20 a + 56\right)\cdot 73^{7} + \left(45 a^{2} + 54 a + 21\right)\cdot 73^{8} + \left(21 a^{2} + 26 a + 6\right)\cdot 73^{9} +O(73^{10})$$ 65*a^2 + 15*a + 30 + (45*a^2 + 46*a + 65)*73 + (22*a^2 + 58*a + 8)*73^2 + (8*a^2 + 22*a + 12)*73^3 + (51*a^2 + 71*a + 62)*73^4 + (39*a^2 + 40*a + 16)*73^5 + (35*a^2 + 67*a + 53)*73^6 + (42*a^2 + 20*a + 56)*73^7 + (45*a^2 + 54*a + 21)*73^8 + (21*a^2 + 26*a + 6)*73^9+O(73^10) $r_{ 3 }$ $=$ $$27 a^{2} + 48 a + 59 + \left(34 a^{2} + 62 a\right)\cdot 73 + \left(66 a^{2} + 28 a + 8\right)\cdot 73^{2} + \left(17 a^{2} + 17 a + 72\right)\cdot 73^{3} + \left(4 a^{2} + 50 a + 66\right)\cdot 73^{4} + \left(66 a^{2} + 28 a + 16\right)\cdot 73^{5} + \left(7 a^{2} + 13 a + 37\right)\cdot 73^{6} + \left(60 a^{2} + 39 a + 16\right)\cdot 73^{7} + \left(34 a + 42\right)\cdot 73^{8} + \left(25 a^{2} + 23 a + 38\right)\cdot 73^{9} +O(73^{10})$$ 27*a^2 + 48*a + 59 + (34*a^2 + 62*a)*73 + (66*a^2 + 28*a + 8)*73^2 + (17*a^2 + 17*a + 72)*73^3 + (4*a^2 + 50*a + 66)*73^4 + (66*a^2 + 28*a + 16)*73^5 + (7*a^2 + 13*a + 37)*73^6 + (60*a^2 + 39*a + 16)*73^7 + (34*a + 42)*73^8 + (25*a^2 + 23*a + 38)*73^9+O(73^10) $r_{ 4 }$ $=$ $$27 a^{2} + 22 + \left(36 a^{2} + 11 a + 16\right)\cdot 73 + \left(20 a^{2} + 15 a + 56\right)\cdot 73^{2} + \left(61 a^{2} + 53 a + 56\right)\cdot 73^{3} + \left(13 a^{2} + 37 a + 11\right)\cdot 73^{4} + \left(63 a^{2} + 51 a + 21\right)\cdot 73^{5} + \left(55 a^{2} + 38 a + 66\right)\cdot 73^{6} + \left(6 a^{2} + 13 a + 23\right)\cdot 73^{7} + \left(24 a^{2} + 37 a + 54\right)\cdot 73^{8} + \left(11 a^{2} + 36 a + 56\right)\cdot 73^{9} +O(73^{10})$$ 27*a^2 + 22 + (36*a^2 + 11*a + 16)*73 + (20*a^2 + 15*a + 56)*73^2 + (61*a^2 + 53*a + 56)*73^3 + (13*a^2 + 37*a + 11)*73^4 + (63*a^2 + 51*a + 21)*73^5 + (55*a^2 + 38*a + 66)*73^6 + (6*a^2 + 13*a + 23)*73^7 + (24*a^2 + 37*a + 54)*73^8 + (11*a^2 + 36*a + 56)*73^9+O(73^10) $r_{ 5 }$ $=$ $$63 a^{2} + 34 + \left(60 a^{2} + 29 a + 60\right)\cdot 73 + \left(4 a^{2} + 56 a + 71\right)\cdot 73^{2} + \left(72 a^{2} + 51 a + 46\right)\cdot 73^{3} + \left(46 a^{2} + 7 a + 26\right)\cdot 73^{4} + \left(38 a^{2} + 72 a + 53\right)\cdot 73^{5} + \left(27 a^{2} + 61 a + 14\right)\cdot 73^{6} + \left(29 a^{2} + 10 a + 24\right)\cdot 73^{7} + \left(59 a^{2} + 37 a + 47\right)\cdot 73^{8} + \left(30 a^{2} + 37 a + 70\right)\cdot 73^{9} +O(73^{10})$$ 63*a^2 + 34 + (60*a^2 + 29*a + 60)*73 + (4*a^2 + 56*a + 71)*73^2 + (72*a^2 + 51*a + 46)*73^3 + (46*a^2 + 7*a + 26)*73^4 + (38*a^2 + 72*a + 53)*73^5 + (27*a^2 + 61*a + 14)*73^6 + (29*a^2 + 10*a + 24)*73^7 + (59*a^2 + 37*a + 47)*73^8 + (30*a^2 + 37*a + 70)*73^9+O(73^10) $r_{ 6 }$ $=$ $$24 a^{2} + 42 a + 18 + \left(62 a^{2} + 53 a + 2\right)\cdot 73 + \left(66 a^{2} + 35 a + 45\right)\cdot 73^{2} + \left(68 a^{2} + 2 a + 42\right)\cdot 73^{3} + \left(16 a^{2} + 57 a + 64\right)\cdot 73^{4} + \left(61 a^{2} + 70 a + 42\right)\cdot 73^{5} + \left(28 a^{2} + 68 a + 54\right)\cdot 73^{6} + \left(15 a^{2} + 23 a + 59\right)\cdot 73^{7} + \left(59 a^{2} + 49 a + 3\right)\cdot 73^{8} + \left(34 a^{2} + 71 a + 15\right)\cdot 73^{9} +O(73^{10})$$ 24*a^2 + 42*a + 18 + (62*a^2 + 53*a + 2)*73 + (66*a^2 + 35*a + 45)*73^2 + (68*a^2 + 2*a + 42)*73^3 + (16*a^2 + 57*a + 64)*73^4 + (61*a^2 + 70*a + 42)*73^5 + (28*a^2 + 68*a + 54)*73^6 + (15*a^2 + 23*a + 59)*73^7 + (59*a^2 + 49*a + 3)*73^8 + (34*a^2 + 71*a + 15)*73^9+O(73^10) $r_{ 7 }$ $=$ $$22 a^{2} + 31 a + 64 + \left(47 a^{2} + 8 a + 30\right)\cdot 73 + \left(58 a^{2} + 22 a + 58\right)\cdot 73^{2} + \left(15 a^{2} + 17 a + 44\right)\cdot 73^{3} + \left(42 a^{2} + 51 a + 49\right)\cdot 73^{4} + \left(21 a^{2} + 23 a + 38\right)\cdot 73^{5} + \left(61 a^{2} + 38 a\right)\cdot 73^{6} + \left(50 a^{2} + 35 a + 34\right)\cdot 73^{7} + \left(62 a^{2} + 59 a + 8\right)\cdot 73^{8} + \left(26 a^{2} + 37 a + 53\right)\cdot 73^{9} +O(73^{10})$$ 22*a^2 + 31*a + 64 + (47*a^2 + 8*a + 30)*73 + (58*a^2 + 22*a + 58)*73^2 + (15*a^2 + 17*a + 44)*73^3 + (42*a^2 + 51*a + 49)*73^4 + (21*a^2 + 23*a + 38)*73^5 + (61*a^2 + 38*a)*73^6 + (50*a^2 + 35*a + 34)*73^7 + (62*a^2 + 59*a + 8)*73^8 + (26*a^2 + 37*a + 53)*73^9+O(73^10) $r_{ 8 }$ $=$ $$56 a^{2} + 25 a + 49 + \left(50 a^{2} + 54 a + 22\right)\cdot 73 + \left(a^{2} + 60 a + 43\right)\cdot 73^{2} + \left(56 a^{2} + 3 a + 25\right)\cdot 73^{3} + \left(21 a^{2} + 15 a + 17\right)\cdot 73^{4} + \left(41 a^{2} + 45 a + 8\right)\cdot 73^{5} + \left(37 a^{2} + 70 a + 28\right)\cdot 73^{6} + \left(56 a^{2} + 22 a + 60\right)\cdot 73^{7} + \left(12 a^{2} + a + 33\right)\cdot 73^{8} + \left(17 a^{2} + 12 a + 52\right)\cdot 73^{9} +O(73^{10})$$ 56*a^2 + 25*a + 49 + (50*a^2 + 54*a + 22)*73 + (a^2 + 60*a + 43)*73^2 + (56*a^2 + 3*a + 25)*73^3 + (21*a^2 + 15*a + 17)*73^4 + (41*a^2 + 45*a + 8)*73^5 + (37*a^2 + 70*a + 28)*73^6 + (56*a^2 + 22*a + 60)*73^7 + (12*a^2 + a + 33)*73^8 + (17*a^2 + 12*a + 52)*73^9+O(73^10) $r_{ 9 }$ $=$ $$45 a^{2} + 42 a + 52 + \left(41 a^{2} + 60 a + 59\right)\cdot 73 + \left(42 a^{2} + 66 a + 59\right)\cdot 73^{2} + \left(58 a^{2} + 10 a + 54\right)\cdot 73^{3} + \left(50 a^{2} + 53 a + 61\right)\cdot 73^{4} + \left(35 a^{2} + 35\right)\cdot 73^{5} + \left(69 a^{2} + 44 a + 25\right)\cdot 73^{6} + \left(48 a^{2} + 27 a + 65\right)\cdot 73^{7} + \left(19 a^{2} + 53 a + 35\right)\cdot 73^{8} + \left(21 a^{2} + 60 a + 54\right)\cdot 73^{9} +O(73^{10})$$ 45*a^2 + 42*a + 52 + (41*a^2 + 60*a + 59)*73 + (42*a^2 + 66*a + 59)*73^2 + (58*a^2 + 10*a + 54)*73^3 + (50*a^2 + 53*a + 61)*73^4 + (35*a^2 + 35)*73^5 + (69*a^2 + 44*a + 25)*73^6 + (48*a^2 + 27*a + 65)*73^7 + (19*a^2 + 53*a + 35)*73^8 + (21*a^2 + 60*a + 54)*73^9+O(73^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.