# Properties

 Label 20.126...992.40t1676.a.a Dimension $20$ Group $(C_2^4:A_5) : C_2$ Conductor $1.263\times 10^{53}$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $20$ Group: $(C_2^4:A_5) : C_2$ Conductor: $$126\!\cdots\!992$$$$\medspace = 2^{40} \cdot 36497^{9}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 10.10.1363999753216.1 Galois orbit size: $1$ Smallest permutation container: 40T1676 Parity: even Determinant: 1.36497.2t1.a.a Projective image: $(C_2^4:A_5) : C_2$ Projective stem field: Galois closure of 10.10.1363999753216.1

## Defining polynomial

 $f(x)$ $=$ $$x^{10} - 9x^{8} + 27x^{6} - 31x^{4} + 12x^{2} - 1$$ x^10 - 9*x^8 + 27*x^6 - 31*x^4 + 12*x^2 - 1 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{4} + 2x^{2} + 15x + 2$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

 Cycle notation $(1,10)(5,6)$ $(1,5,4,2,3)(6,10,9,7,8)$ $(1,10,9,7,8)(2,3,6,5,4)$ $(1,5)(6,10)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character value $1$ $1$ $()$ $20$ $5$ $2$ $(1,6)(2,7)(3,8)(4,9)$ $4$ $10$ $2$ $(1,6)(5,10)$ $-4$ $20$ $2$ $(1,10)(5,6)$ $2$ $60$ $2$ $(2,5)(3,4)(7,10)(8,9)$ $0$ $60$ $2$ $(1,5)(2,7)(4,9)(6,10)$ $2$ $80$ $3$ $(1,2,5)(6,7,10)$ $-1$ $20$ $4$ $(1,4,6,9)(2,7)(3,8)(5,10)$ $-2$ $60$ $4$ $(1,7,6,2)(3,9,8,4)$ $0$ $60$ $4$ $(1,3,6,8)(2,7)$ $-2$ $120$ $4$ $(1,6)(2,10,7,5)(3,4)(8,9)$ $0$ $240$ $4$ $(2,3,5,4)(7,8,10,9)$ $0$ $384$ $5$ $(1,5,4,2,3)(6,10,9,7,8)$ $0$ $80$ $6$ $(1,6)(2,8,5)(3,10,7)(4,9)$ $-1$ $160$ $6$ $(1,5,2)(3,4)(6,10,7)(8,9)$ $-1$ $160$ $6$ $(1,4,10,6,9,5)(2,7)$ $1$ $240$ $8$ $(1,9,7,8,6,4,2,3)(5,10)$ $0$ $160$ $12$ $(1,9,6,4)(2,10,8,7,5,3)$ $1$

The blue line marks the conjugacy class containing complex conjugation.