# Properties

 Label 12.218...641.18t206.a.a Dimension $12$ Group $S_3 \wr C_3$ Conductor $2.188\times 10^{17}$ Root number $1$ Indicator $1$

# Learn more

## Basic invariants

 Dimension: $12$ Group: $S_3 \wr C_3$ Conductor: $$218768667202582641$$$$\medspace = 3^{20} \cdot 89^{4}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.5.3831158169.1 Galois orbit size: $1$ Smallest permutation container: 18T206 Parity: even Determinant: 1.1.1t1.a.a Projective image: $C_3^3:C_2^2.C_6$ Projective stem field: Galois closure of 9.5.3831158169.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $$x^{3} + 3x + 51$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.