# Properties

 Label 12.118...000.24t2821.a.a Dimension $12$ Group $S_4\wr C_2$ Conductor $1.186\times 10^{17}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $12$ Group: $S_4\wr C_2$ Conductor: $$118636749824000000$$$$\medspace = 2^{16} \cdot 5^{6} \cdot 41^{5}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 8.4.11027360000.1 Galois orbit size: $1$ Smallest permutation container: 24T2821 Parity: even Determinant: 1.41.2t1.a.a Projective image: $S_4\wr C_2$ Projective stem field: Galois closure of 8.4.11027360000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{8} - 2x^{7} + 6x^{6} - 2x^{5} + 26x^{4} - 24x^{3} - 24x^{2} + 16x + 4$$ x^8 - 2*x^7 + 6*x^6 - 2*x^5 + 26*x^4 - 24*x^3 - 24*x^2 + 16*x + 4 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{4} + 3x^{2} + 16x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$13 + 10\cdot 31 + 16\cdot 31^{2} + 21\cdot 31^{3} + 16\cdot 31^{4} + 26\cdot 31^{5} + 27\cdot 31^{6} + 7\cdot 31^{7} + 15\cdot 31^{8} + 26\cdot 31^{9} +O(31^{10})$$ 13 + 10*31 + 16*31^2 + 21*31^3 + 16*31^4 + 26*31^5 + 27*31^6 + 7*31^7 + 15*31^8 + 26*31^9+O(31^10) $r_{ 2 }$ $=$ $$16 a^{3} + 16 a^{2} + 26 a + 21 + \left(16 a^{3} + 13 a^{2} + 21 a + 4\right)\cdot 31 + \left(23 a^{3} + 9 a^{2} + 2 a + 8\right)\cdot 31^{2} + \left(25 a^{3} + 2 a^{2} + 12 a + 18\right)\cdot 31^{3} + \left(12 a^{3} + 7 a^{2} + 12 a + 24\right)\cdot 31^{4} + \left(26 a^{3} + 9 a^{2} + 11 a + 9\right)\cdot 31^{5} + \left(10 a^{3} + 14 a^{2} + 16 a + 17\right)\cdot 31^{6} + \left(7 a^{3} + a^{2} + 7 a + 13\right)\cdot 31^{7} + \left(24 a^{3} + 10 a\right)\cdot 31^{8} + \left(14 a^{3} + 21 a^{2} + 13 a + 25\right)\cdot 31^{9} +O(31^{10})$$ 16*a^3 + 16*a^2 + 26*a + 21 + (16*a^3 + 13*a^2 + 21*a + 4)*31 + (23*a^3 + 9*a^2 + 2*a + 8)*31^2 + (25*a^3 + 2*a^2 + 12*a + 18)*31^3 + (12*a^3 + 7*a^2 + 12*a + 24)*31^4 + (26*a^3 + 9*a^2 + 11*a + 9)*31^5 + (10*a^3 + 14*a^2 + 16*a + 17)*31^6 + (7*a^3 + a^2 + 7*a + 13)*31^7 + (24*a^3 + 10*a)*31^8 + (14*a^3 + 21*a^2 + 13*a + 25)*31^9+O(31^10) $r_{ 3 }$ $=$ $$2 a^{3} + 15 a^{2} + 4 a + 19 + \left(17 a^{3} + 8 a^{2} + 14 a\right)\cdot 31 + \left(5 a^{3} + 3 a^{2} + 30 a\right)\cdot 31^{2} + \left(21 a^{3} + 12 a^{2} + 10 a + 13\right)\cdot 31^{3} + \left(16 a^{3} + 8 a^{2} + 23 a + 18\right)\cdot 31^{4} + \left(22 a^{3} + 30 a^{2} + 10 a + 30\right)\cdot 31^{5} + \left(22 a^{3} + 15 a^{2} + 9 a + 30\right)\cdot 31^{6} + \left(23 a^{3} + 8 a^{2} + 24\right)\cdot 31^{7} + \left(21 a^{3} + 7 a^{2} + 16\right)\cdot 31^{8} + \left(23 a^{3} + a^{2} + 20 a + 15\right)\cdot 31^{9} +O(31^{10})$$ 2*a^3 + 15*a^2 + 4*a + 19 + (17*a^3 + 8*a^2 + 14*a)*31 + (5*a^3 + 3*a^2 + 30*a)*31^2 + (21*a^3 + 12*a^2 + 10*a + 13)*31^3 + (16*a^3 + 8*a^2 + 23*a + 18)*31^4 + (22*a^3 + 30*a^2 + 10*a + 30)*31^5 + (22*a^3 + 15*a^2 + 9*a + 30)*31^6 + (23*a^3 + 8*a^2 + 24)*31^7 + (21*a^3 + 7*a^2 + 16)*31^8 + (23*a^3 + a^2 + 20*a + 15)*31^9+O(31^10) $r_{ 4 }$ $=$ $$29 a^{3} + 16 a^{2} + 27 a + 19 + \left(13 a^{3} + 22 a^{2} + 16 a + 30\right)\cdot 31 + \left(25 a^{3} + 27 a^{2} + 11\right)\cdot 31^{2} + \left(9 a^{3} + 18 a^{2} + 20 a + 26\right)\cdot 31^{3} + \left(14 a^{3} + 22 a^{2} + 7 a + 26\right)\cdot 31^{4} + \left(8 a^{3} + 20 a + 18\right)\cdot 31^{5} + \left(8 a^{3} + 15 a^{2} + 21 a + 26\right)\cdot 31^{6} + \left(7 a^{3} + 22 a^{2} + 30 a + 18\right)\cdot 31^{7} + \left(9 a^{3} + 23 a^{2} + 30 a + 30\right)\cdot 31^{8} + \left(7 a^{3} + 29 a^{2} + 10 a\right)\cdot 31^{9} +O(31^{10})$$ 29*a^3 + 16*a^2 + 27*a + 19 + (13*a^3 + 22*a^2 + 16*a + 30)*31 + (25*a^3 + 27*a^2 + 11)*31^2 + (9*a^3 + 18*a^2 + 20*a + 26)*31^3 + (14*a^3 + 22*a^2 + 7*a + 26)*31^4 + (8*a^3 + 20*a + 18)*31^5 + (8*a^3 + 15*a^2 + 21*a + 26)*31^6 + (7*a^3 + 22*a^2 + 30*a + 18)*31^7 + (9*a^3 + 23*a^2 + 30*a + 30)*31^8 + (7*a^3 + 29*a^2 + 10*a)*31^9+O(31^10) $r_{ 5 }$ $=$ $$25 a^{3} + 28 a^{2} + 25 a + 23 + \left(a^{3} + 6 a^{2} + 7 a + 19\right)\cdot 31 + \left(29 a^{3} + 19 a^{2} + a + 11\right)\cdot 31^{2} + \left(13 a^{3} + 22 a^{2} + 24 a + 15\right)\cdot 31^{3} + \left(4 a^{3} + 16 a^{2} + 26 a\right)\cdot 31^{4} + \left(8 a^{3} + 14 a^{2} + 3 a\right)\cdot 31^{5} + \left(13 a^{3} + 24 a^{2} + 9 a + 15\right)\cdot 31^{6} + \left(16 a^{3} + 10 a^{2} + 14 a + 12\right)\cdot 31^{7} + \left(18 a^{3} + 23 a^{2} + 5 a + 13\right)\cdot 31^{8} + \left(12 a^{3} + 27 a^{2} + 6 a + 24\right)\cdot 31^{9} +O(31^{10})$$ 25*a^3 + 28*a^2 + 25*a + 23 + (a^3 + 6*a^2 + 7*a + 19)*31 + (29*a^3 + 19*a^2 + a + 11)*31^2 + (13*a^3 + 22*a^2 + 24*a + 15)*31^3 + (4*a^3 + 16*a^2 + 26*a)*31^4 + (8*a^3 + 14*a^2 + 3*a)*31^5 + (13*a^3 + 24*a^2 + 9*a + 15)*31^6 + (16*a^3 + 10*a^2 + 14*a + 12)*31^7 + (18*a^3 + 23*a^2 + 5*a + 13)*31^8 + (12*a^3 + 27*a^2 + 6*a + 24)*31^9+O(31^10) $r_{ 6 }$ $=$ $$12 a^{3} + 9 a^{2} + 27 a + 9 + \left(28 a^{3} + 4 a^{2} + 26 a + 9\right)\cdot 31 + \left(21 a^{3} + 28 a^{2} + 17 a + 1\right)\cdot 31^{2} + \left(7 a^{3} + 29 a^{2} + 30 a + 29\right)\cdot 31^{3} + \left(3 a^{3} + 24 a^{2} + 12 a + 13\right)\cdot 31^{4} + \left(9 a^{3} + 21 a^{2} + 16 a + 22\right)\cdot 31^{5} + \left(9 a^{3} + 9 a^{2} + 6 a + 22\right)\cdot 31^{6} + \left(16 a^{3} + 11 a^{2} + 27 a + 11\right)\cdot 31^{7} + \left(16 a^{3} + 24 a^{2} + 7 a + 6\right)\cdot 31^{8} + \left(2 a^{3} + 27 a^{2} + 10 a + 12\right)\cdot 31^{9} +O(31^{10})$$ 12*a^3 + 9*a^2 + 27*a + 9 + (28*a^3 + 4*a^2 + 26*a + 9)*31 + (21*a^3 + 28*a^2 + 17*a + 1)*31^2 + (7*a^3 + 29*a^2 + 30*a + 29)*31^3 + (3*a^3 + 24*a^2 + 12*a + 13)*31^4 + (9*a^3 + 21*a^2 + 16*a + 22)*31^5 + (9*a^3 + 9*a^2 + 6*a + 22)*31^6 + (16*a^3 + 11*a^2 + 27*a + 11)*31^7 + (16*a^3 + 24*a^2 + 7*a + 6)*31^8 + (2*a^3 + 27*a^2 + 10*a + 12)*31^9+O(31^10) $r_{ 7 }$ $=$ $$18 + 7\cdot 31 + 9\cdot 31^{2} + 2\cdot 31^{4} + 30\cdot 31^{5} + 17\cdot 31^{6} + 7\cdot 31^{7} + 14\cdot 31^{8} + 10\cdot 31^{9} +O(31^{10})$$ 18 + 7*31 + 9*31^2 + 2*31^4 + 30*31^5 + 17*31^6 + 7*31^7 + 14*31^8 + 10*31^9+O(31^10) $r_{ 8 }$ $=$ $$9 a^{3} + 9 a^{2} + 15 a + 4 + \left(15 a^{3} + 6 a^{2} + 5 a + 10\right)\cdot 31 + \left(18 a^{3} + 5 a^{2} + 9 a + 3\right)\cdot 31^{2} + \left(14 a^{3} + 7 a^{2} + 26 a\right)\cdot 31^{3} + \left(10 a^{3} + 13 a^{2} + 9 a + 21\right)\cdot 31^{4} + \left(18 a^{3} + 16 a^{2} + 30 a + 16\right)\cdot 31^{5} + \left(28 a^{3} + 13 a^{2} + 29 a + 27\right)\cdot 31^{6} + \left(21 a^{3} + 7 a^{2} + 12 a + 26\right)\cdot 31^{7} + \left(2 a^{3} + 14 a^{2} + 7 a + 26\right)\cdot 31^{8} + \left(a^{3} + 16 a^{2} + a + 8\right)\cdot 31^{9} +O(31^{10})$$ 9*a^3 + 9*a^2 + 15*a + 4 + (15*a^3 + 6*a^2 + 5*a + 10)*31 + (18*a^3 + 5*a^2 + 9*a + 3)*31^2 + (14*a^3 + 7*a^2 + 26*a)*31^3 + (10*a^3 + 13*a^2 + 9*a + 21)*31^4 + (18*a^3 + 16*a^2 + 30*a + 16)*31^5 + (28*a^3 + 13*a^2 + 29*a + 27)*31^6 + (21*a^3 + 7*a^2 + 12*a + 26)*31^7 + (2*a^3 + 14*a^2 + 7*a + 26)*31^8 + (a^3 + 16*a^2 + a + 8)*31^9+O(31^10)

## Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $12$ $6$ $2$ $(1,4)(3,7)$ $4$ $9$ $2$ $(1,4)(2,6)(3,7)(5,8)$ $-4$ $12$ $2$ $(2,5)$ $2$ $24$ $2$ $(1,2)(3,5)(4,6)(7,8)$ $0$ $36$ $2$ $(1,3)(2,5)$ $0$ $36$ $2$ $(1,4)(2,5)(3,7)$ $2$ $16$ $3$ $(2,6,8)$ $-3$ $64$ $3$ $(2,6,8)(3,4,7)$ $0$ $12$ $4$ $(1,3,4,7)$ $-2$ $36$ $4$ $(1,3,4,7)(2,5,6,8)$ $0$ $36$ $4$ $(1,4)(2,5,6,8)(3,7)$ $-2$ $72$ $4$ $(1,6,4,2)(3,8,7,5)$ $0$ $72$ $4$ $(1,3,4,7)(2,5)$ $0$ $144$ $4$ $(1,2,3,5)(4,6)(7,8)$ $0$ $48$ $6$ $(1,4)(2,8,6)(3,7)$ $1$ $96$ $6$ $(2,5)(3,7,4)$ $-1$ $192$ $6$ $(1,5)(2,3,6,4,8,7)$ $0$ $144$ $8$ $(1,5,3,6,4,8,7,2)$ $0$ $96$ $12$ $(1,3,4,7)(2,6,8)$ $1$

The blue line marks the conjugacy class containing complex conjugation.