# Properties

 Label 12.270...625.18t206.a.a Dimension $12$ Group $S_3 \wr C_3$ Conductor $2.709\times 10^{17}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $12$ Group: $S_3 \wr C_3$ Conductor: $$270945468811850625$$$$\medspace = 3^{12} \cdot 5^{4} \cdot 13^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.5.12825821008845.1 Galois orbit size: $1$ Smallest permutation container: 18T206 Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_3\wr C_3$ Projective stem field: Galois closure of 9.5.12825821008845.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{7} - 17x^{6} - 36x^{5} + 21x^{4} + 39x^{3} + 9x^{2} + 90x + 53$$ x^9 - 3*x^7 - 17*x^6 - 36*x^5 + 21*x^4 + 39*x^3 + 9*x^2 + 90*x + 53 .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{3} + 2x + 27$$

Roots:
 $r_{ 1 }$ $=$ $$20 a^{2} + 9 a + 23 + \left(5 a + 24\right)\cdot 29 + \left(25 a^{2} + 24 a + 9\right)\cdot 29^{2} + \left(11 a^{2} + 25 a + 1\right)\cdot 29^{3} + \left(25 a^{2} + 18 a + 19\right)\cdot 29^{4} + \left(26 a^{2} + 2 a + 7\right)\cdot 29^{5} + \left(a^{2} + 4\right)\cdot 29^{6} + \left(14 a^{2} + 3 a + 19\right)\cdot 29^{7} + \left(28 a^{2} + a + 11\right)\cdot 29^{8} + \left(16 a^{2} + 19 a + 10\right)\cdot 29^{9} +O(29^{10})$$ 20*a^2 + 9*a + 23 + (5*a + 24)*29 + (25*a^2 + 24*a + 9)*29^2 + (11*a^2 + 25*a + 1)*29^3 + (25*a^2 + 18*a + 19)*29^4 + (26*a^2 + 2*a + 7)*29^5 + (a^2 + 4)*29^6 + (14*a^2 + 3*a + 19)*29^7 + (28*a^2 + a + 11)*29^8 + (16*a^2 + 19*a + 10)*29^9+O(29^10) $r_{ 2 }$ $=$ $$14 a^{2} + 26 + \left(2 a^{2} + 6 a + 23\right)\cdot 29 + \left(25 a^{2} + 8 a + 19\right)\cdot 29^{2} + \left(13 a^{2} + 7 a + 1\right)\cdot 29^{3} + \left(7 a^{2} + 16 a + 26\right)\cdot 29^{4} + \left(23 a^{2} + 23 a + 28\right)\cdot 29^{5} + \left(13 a^{2} + 28 a + 5\right)\cdot 29^{6} + \left(a^{2} + 21 a + 27\right)\cdot 29^{7} + \left(15 a^{2} + 8 a + 4\right)\cdot 29^{8} + \left(6 a^{2} + 21 a + 10\right)\cdot 29^{9} +O(29^{10})$$ 14*a^2 + 26 + (2*a^2 + 6*a + 23)*29 + (25*a^2 + 8*a + 19)*29^2 + (13*a^2 + 7*a + 1)*29^3 + (7*a^2 + 16*a + 26)*29^4 + (23*a^2 + 23*a + 28)*29^5 + (13*a^2 + 28*a + 5)*29^6 + (a^2 + 21*a + 27)*29^7 + (15*a^2 + 8*a + 4)*29^8 + (6*a^2 + 21*a + 10)*29^9+O(29^10) $r_{ 3 }$ $=$ $$2 a^{2} + 15 a + 28 + \left(8 a^{2} + 28 a + 4\right)\cdot 29 + \left(6 a^{2} + 26 a + 26\right)\cdot 29^{2} + \left(15 a^{2} + 27 a + 12\right)\cdot 29^{3} + \left(27 a^{2} + 19 a + 6\right)\cdot 29^{4} + \left(5 a^{2} + 3 a + 9\right)\cdot 29^{5} + \left(27 a^{2} + 6 a + 8\right)\cdot 29^{6} + \left(12 a^{2} + 21 a + 1\right)\cdot 29^{7} + \left(27 a^{2} + 6 a + 20\right)\cdot 29^{8} + \left(24 a^{2} + 26 a + 24\right)\cdot 29^{9} +O(29^{10})$$ 2*a^2 + 15*a + 28 + (8*a^2 + 28*a + 4)*29 + (6*a^2 + 26*a + 26)*29^2 + (15*a^2 + 27*a + 12)*29^3 + (27*a^2 + 19*a + 6)*29^4 + (5*a^2 + 3*a + 9)*29^5 + (27*a^2 + 6*a + 8)*29^6 + (12*a^2 + 21*a + 1)*29^7 + (27*a^2 + 6*a + 20)*29^8 + (24*a^2 + 26*a + 24)*29^9+O(29^10) $r_{ 4 }$ $=$ $$3 a^{2} + 23 a + 10 + \left(24 a^{2} + 11 a + 26\right)\cdot 29 + \left(14 a^{2} + 5 a + 27\right)\cdot 29^{2} + \left(10 a^{2} + 5 a + 25\right)\cdot 29^{3} + \left(5 a^{2} + 15 a + 5\right)\cdot 29^{4} + \left(15 a^{2} + 22 a + 2\right)\cdot 29^{5} + \left(17 a^{2} + 22 a + 5\right)\cdot 29^{6} + \left(2 a^{2} + 3 a + 26\right)\cdot 29^{7} + \left(16 a^{2} + 3 a + 4\right)\cdot 29^{8} + \left(27 a^{2} + 14 a + 28\right)\cdot 29^{9} +O(29^{10})$$ 3*a^2 + 23*a + 10 + (24*a^2 + 11*a + 26)*29 + (14*a^2 + 5*a + 27)*29^2 + (10*a^2 + 5*a + 25)*29^3 + (5*a^2 + 15*a + 5)*29^4 + (15*a^2 + 22*a + 2)*29^5 + (17*a^2 + 22*a + 5)*29^6 + (2*a^2 + 3*a + 26)*29^7 + (16*a^2 + 3*a + 4)*29^8 + (27*a^2 + 14*a + 28)*29^9+O(29^10) $r_{ 5 }$ $=$ $$13 a + 17 + \left(20 a^{2} + a + 8\right)\cdot 29 + \left(24 a^{2} + 8 a + 19\right)\cdot 29^{2} + \left(22 a^{2} + 6 a + 13\right)\cdot 29^{3} + \left(22 a^{2} + 13 a + 17\right)\cdot 29^{4} + \left(5 a^{2} + 3 a + 5\right)\cdot 29^{5} + \left(20 a^{2} + 9 a + 24\right)\cdot 29^{6} + \left(14 a^{2} + 15\right)\cdot 29^{7} + \left(24 a^{2} + 12 a + 17\right)\cdot 29^{8} + \left(4 a^{2} + 6 a + 17\right)\cdot 29^{9} +O(29^{10})$$ 13*a + 17 + (20*a^2 + a + 8)*29 + (24*a^2 + 8*a + 19)*29^2 + (22*a^2 + 6*a + 13)*29^3 + (22*a^2 + 13*a + 17)*29^4 + (5*a^2 + 3*a + 5)*29^5 + (20*a^2 + 9*a + 24)*29^6 + (14*a^2 + 15)*29^7 + (24*a^2 + 12*a + 17)*29^8 + (4*a^2 + 6*a + 17)*29^9+O(29^10) $r_{ 6 }$ $=$ $$15 a^{2} + 16 a + 8 + \left(6 a^{2} + 21 a\right)\cdot 29 + \left(8 a^{2} + 12 a + 7\right)\cdot 29^{2} + \left(21 a^{2} + 15 a + 21\right)\cdot 29^{3} + \left(27 a^{2} + 28 a + 4\right)\cdot 29^{4} + \left(28 a^{2} + a + 17\right)\cdot 29^{5} + \left(23 a^{2} + 20 a + 19\right)\cdot 29^{6} + \left(12 a^{2} + 6 a + 3\right)\cdot 29^{7} + \left(18 a^{2} + 8 a + 19\right)\cdot 29^{8} + \left(17 a^{2} + a + 5\right)\cdot 29^{9} +O(29^{10})$$ 15*a^2 + 16*a + 8 + (6*a^2 + 21*a)*29 + (8*a^2 + 12*a + 7)*29^2 + (21*a^2 + 15*a + 21)*29^3 + (27*a^2 + 28*a + 4)*29^4 + (28*a^2 + a + 17)*29^5 + (23*a^2 + 20*a + 19)*29^6 + (12*a^2 + 6*a + 3)*29^7 + (18*a^2 + 8*a + 19)*29^8 + (17*a^2 + a + 5)*29^9+O(29^10) $r_{ 7 }$ $=$ $$12 a^{2} + 27 a + 22 + \left(14 a^{2} + 7 a + 23\right)\cdot 29 + \left(14 a^{2} + 18 a + 24\right)\cdot 29^{2} + \left(21 a^{2} + 14 a + 23\right)\cdot 29^{3} + \left(2 a^{2} + 9 a + 17\right)\cdot 29^{4} + \left(23 a^{2} + 23 a + 2\right)\cdot 29^{5} + \left(6 a^{2} + 2 a + 1\right)\cdot 29^{6} + \left(3 a^{2} + a + 24\right)\cdot 29^{7} + \left(12 a^{2} + 14 a + 18\right)\cdot 29^{8} + \left(15 a^{2} + a + 27\right)\cdot 29^{9} +O(29^{10})$$ 12*a^2 + 27*a + 22 + (14*a^2 + 7*a + 23)*29 + (14*a^2 + 18*a + 24)*29^2 + (21*a^2 + 14*a + 23)*29^3 + (2*a^2 + 9*a + 17)*29^4 + (23*a^2 + 23*a + 2)*29^5 + (6*a^2 + 2*a + 1)*29^6 + (3*a^2 + a + 24)*29^7 + (12*a^2 + 14*a + 18)*29^8 + (15*a^2 + a + 27)*29^9+O(29^10) $r_{ 8 }$ $=$ $$26 a^{2} + 22 a + 2 + \left(13 a^{2} + 15 a + 23\right)\cdot 29 + \left(18 a^{2} + 15 a + 10\right)\cdot 29^{2} + \left(24 a^{2} + 17 a + 18\right)\cdot 29^{3} + 5\cdot 29^{4} + \left(8 a^{2} + 3 a + 21\right)\cdot 29^{5} + \left(20 a^{2} + 26 a + 28\right)\cdot 29^{6} + \left(11 a^{2} + 24 a + 15\right)\cdot 29^{7} + \left(17 a^{2} + 13 a + 6\right)\cdot 29^{8} + \left(25 a^{2} + 8 a + 12\right)\cdot 29^{9} +O(29^{10})$$ 26*a^2 + 22*a + 2 + (13*a^2 + 15*a + 23)*29 + (18*a^2 + 15*a + 10)*29^2 + (24*a^2 + 17*a + 18)*29^3 + 5*29^4 + (8*a^2 + 3*a + 21)*29^5 + (20*a^2 + 26*a + 28)*29^6 + (11*a^2 + 24*a + 15)*29^7 + (17*a^2 + 13*a + 6)*29^8 + (25*a^2 + 8*a + 12)*29^9+O(29^10) $r_{ 9 }$ $=$ $$24 a^{2} + 20 a + 9 + \left(25 a^{2} + 17 a + 9\right)\cdot 29 + \left(7 a^{2} + 25 a + 28\right)\cdot 29^{2} + \left(3 a^{2} + 24 a + 25\right)\cdot 29^{3} + \left(25 a^{2} + 22 a + 12\right)\cdot 29^{4} + \left(7 a^{2} + 2 a + 21\right)\cdot 29^{5} + \left(13 a^{2} + 18\right)\cdot 29^{6} + \left(13 a^{2} + 4 a + 11\right)\cdot 29^{7} + \left(14 a^{2} + 19 a + 12\right)\cdot 29^{8} + \left(5 a^{2} + 17 a + 8\right)\cdot 29^{9} +O(29^{10})$$ 24*a^2 + 20*a + 9 + (25*a^2 + 17*a + 9)*29 + (7*a^2 + 25*a + 28)*29^2 + (3*a^2 + 24*a + 25)*29^3 + (25*a^2 + 22*a + 12)*29^4 + (7*a^2 + 2*a + 21)*29^5 + (13*a^2 + 18)*29^6 + (13*a^2 + 4*a + 11)*29^7 + (14*a^2 + 19*a + 12)*29^8 + (5*a^2 + 17*a + 8)*29^9+O(29^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.