# Properties

 Label 30.275...000.144.a.b Dimension $30$ Group $A_5 \wr C_2$ Conductor $2.759\times 10^{143}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $30$ Group: $A_5 \wr C_2$ Conductor: $$275\!\cdots\!000$$$$\medspace = 2^{69} \cdot 5^{16} \cdot 71^{22} \cdot 26449^{16}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 10.10.23300317255158046392320000.1 Galois orbit size: $2$ Smallest permutation container: 144 Parity: even Determinant: 1.8.2t1.a.a Projective image: $A_5 \wr C_2$ Projective stem field: Galois closure of 10.10.23300317255158046392320000.1

## Defining polynomial

 $f(x)$ $=$ $$x^{10} - 2 x^{9} - 345 x^{8} + 720 x^{7} + 33292 x^{6} - 77348 x^{5} - 795480 x^{4} + 1563624 x^{3} + 4991153 x^{2} - 5404190 x - 10994849$$ x^10 - 2*x^9 - 345*x^8 + 720*x^7 + 33292*x^6 - 77348*x^5 - 795480*x^4 + 1563624*x^3 + 4991153*x^2 - 5404190*x - 10994849 .

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

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 10 }$ Character value $1$ $1$ $()$ $30$ $30$ $2$ $(4,5)(7,9)$ $-2$ $60$ $2$ $(1,4)(2,5)(3,10)(6,9)(7,8)$ $0$ $225$ $2$ $(1,2)(4,5)(6,8)(7,9)$ $-2$ $40$ $3$ $(4,5,7)$ $-3$ $400$ $3$ $(1,2,8)(4,5,7)$ $0$ $900$ $4$ $(1,7,8,4)(2,10,3,5)(6,9)$ $0$ $24$ $5$ $(4,5,7,9,10)$ $-5 \zeta_{5}^{3} - 5 \zeta_{5}^{2}$ $24$ $5$ $(4,7,9,10,5)$ $5 \zeta_{5}^{3} + 5 \zeta_{5}^{2} + 5$ $144$ $5$ $(1,2,8,6,3)(4,5,7,9,10)$ $0$ $144$ $5$ $(1,8,6,3,2)(4,7,9,10,5)$ $0$ $288$ $5$ $(1,2,8,6,3)(4,7,9,10,5)$ $0$ $600$ $6$ $(1,2)(4,5,7)(6,8)$ $1$ $1200$ $6$ $(1,7,8,5,2,4)(3,10)(6,9)$ $0$ $360$ $10$ $(1,2)(4,5,7,9,10)(6,8)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $360$ $10$ $(1,2)(4,7,9,10,5)(6,8)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $720$ $10$ $(1,7,8,5,2,10,3,9,6,4)$ $0$ $720$ $10$ $(1,5,2,10,3,9,6,7,8,4)$ $0$ $480$ $15$ $(1,2,8)(4,5,7,9,10)$ $\zeta_{5}^{3} + \zeta_{5}^{2}$ $480$ $15$ $(1,2,8)(4,7,9,10,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$

The blue line marks the conjugacy class containing complex conjugation.