Basic invariants
Dimension: | $6$ |
Group: | $(A_4\wr C_2):C_2$ |
Conductor: | \(59270400\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 5^{2} \cdot 7^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.4.1244678400.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $(A_4\wr C_2):C_2$ |
Parity: | even |
Determinant: | 1.21.2t1.a.a |
Projective image: | $\PGOPlus(4,3)$ |
Projective stem field: | Galois closure of 8.4.1244678400.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 8x^{5} + 9x^{4} - 10x^{3} - 10x^{2} + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{3} + 6x + 35 \)
Roots:
$r_{ 1 }$ | $=$ | \( 18 a^{2} + 3 a + 27 + \left(a + 26\right)\cdot 37 + \left(14 a^{2} + a + 35\right)\cdot 37^{2} + \left(31 a^{2} + 22 a + 15\right)\cdot 37^{3} + \left(3 a^{2} + 11 a + 2\right)\cdot 37^{4} + \left(5 a^{2} + 24 a + 24\right)\cdot 37^{5} + \left(20 a^{2} + 18 a + 32\right)\cdot 37^{6} + \left(12 a^{2} + 5 a + 4\right)\cdot 37^{7} + \left(29 a^{2} + 15 a + 26\right)\cdot 37^{8} + \left(7 a^{2} + 16 a + 1\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 2 }$ | $=$ | \( 31 + 28\cdot 37 + 16\cdot 37^{2} + 13\cdot 37^{3} + 17\cdot 37^{4} + 34\cdot 37^{5} + 37^{6} + 23\cdot 37^{7} + 21\cdot 37^{8} + 28\cdot 37^{9} +O(37^{10})\) |
$r_{ 3 }$ | $=$ | \( 26 + 36\cdot 37 + 23\cdot 37^{2} + 32\cdot 37^{3} + 37^{4} + 26\cdot 37^{5} + 32\cdot 37^{6} + 24\cdot 37^{7} + 14\cdot 37^{8} + 14\cdot 37^{9} +O(37^{10})\) |
$r_{ 4 }$ | $=$ | \( 6 a^{2} + 31 a + 2 + \left(15 a^{2} + 31 a + 14\right)\cdot 37 + \left(10 a^{2} + 29 a + 11\right)\cdot 37^{2} + \left(36 a^{2} + 23 a + 17\right)\cdot 37^{3} + \left(13 a^{2} + 24 a + 25\right)\cdot 37^{4} + \left(5 a^{2} + 3 a + 34\right)\cdot 37^{5} + \left(2 a^{2} + 9 a + 7\right)\cdot 37^{6} + \left(6 a^{2} + 4 a + 4\right)\cdot 37^{7} + \left(36 a^{2} + 5 a + 14\right)\cdot 37^{8} + \left(30 a^{2} + 36 a + 3\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 5 }$ | $=$ | \( 23 a^{2} + 2 a + 10 + \left(18 a^{2} + 14 a + 25\right)\cdot 37 + \left(30 a^{2} + 7 a + 27\right)\cdot 37^{2} + \left(23 a^{2} + 22 a + 22\right)\cdot 37^{3} + \left(20 a^{2} + 18 a + 32\right)\cdot 37^{4} + \left(36 a^{2} + 30 a + 1\right)\cdot 37^{5} + \left(13 a^{2} + 18 a + 8\right)\cdot 37^{6} + \left(9 a^{2} + 11 a + 29\right)\cdot 37^{7} + \left(5 a^{2} + 36 a + 3\right)\cdot 37^{8} + \left(8 a^{2} + 27 a + 3\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 6 }$ | $=$ | \( 33 a^{2} + 32 a + 13 + \left(17 a^{2} + 21 a + 22\right)\cdot 37 + \left(29 a^{2} + 28 a + 23\right)\cdot 37^{2} + \left(18 a^{2} + 29 a + 2\right)\cdot 37^{3} + \left(12 a^{2} + 6 a\right)\cdot 37^{4} + \left(32 a^{2} + 19 a + 22\right)\cdot 37^{5} + \left(2 a^{2} + 36 a\right)\cdot 37^{6} + \left(15 a^{2} + 19 a + 15\right)\cdot 37^{7} + \left(2 a^{2} + 22 a + 29\right)\cdot 37^{8} + \left(21 a^{2} + 29 a + 17\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 7 }$ | $=$ | \( 23 a^{2} + 14 a + 33 + \left(15 a^{2} + a + 15\right)\cdot 37 + \left(29 a^{2} + 10 a + 13\right)\cdot 37^{2} + \left(6 a^{2} + 27 a + 10\right)\cdot 37^{3} + \left(6 a^{2} + 5 a + 31\right)\cdot 37^{4} + \left(33 a^{2} + 6 a + 34\right)\cdot 37^{5} + \left(25 a^{2} + 23 a + 28\right)\cdot 37^{6} + \left(19 a^{2} + 24 a + 21\right)\cdot 37^{7} + \left(35 a^{2} + 6 a + 11\right)\cdot 37^{8} + \left(5 a^{2} + 35 a + 14\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 8 }$ | $=$ | \( 8 a^{2} + 29 a + 10 + \left(6 a^{2} + 3 a + 15\right)\cdot 37 + \left(34 a^{2} + 34 a + 32\right)\cdot 37^{2} + \left(30 a^{2} + 22 a + 32\right)\cdot 37^{3} + \left(16 a^{2} + 6 a + 36\right)\cdot 37^{4} + \left(35 a^{2} + 27 a + 6\right)\cdot 37^{5} + \left(8 a^{2} + 4 a + 35\right)\cdot 37^{6} + \left(11 a^{2} + 8 a + 24\right)\cdot 37^{7} + \left(2 a^{2} + 25 a + 26\right)\cdot 37^{8} + \left(2 a + 27\right)\cdot 37^{9} +O(37^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $6$ |
$6$ | $2$ | $(2,7)(4,8)$ | $2$ |
$9$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $-2$ |
$12$ | $2$ | $(1,8)(2,3)(4,5)(6,7)$ | $0$ |
$12$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
$36$ | $2$ | $(3,5)(7,8)$ | $2$ |
$16$ | $3$ | $(1,6,3)$ | $3$ |
$32$ | $3$ | $(1,6,3)(2,8,7)$ | $0$ |
$32$ | $3$ | $(1,6,3)(2,8,4)$ | $0$ |
$36$ | $4$ | $(1,8,5,4)(2,6,7,3)$ | $0$ |
$36$ | $4$ | $(1,6,5,3)(2,4,7,8)$ | $-2$ |
$36$ | $4$ | $(1,2,5,7)(3,4,6,8)$ | $0$ |
$72$ | $4$ | $(2,8,7,4)(5,6)$ | $0$ |
$48$ | $6$ | $(1,6,3)(2,7)(4,8)$ | $-1$ |
$96$ | $6$ | $(1,7,6,2,3,8)(4,5)$ | $0$ |
$96$ | $6$ | $(1,8,6,4,3,2)(5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.