Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(123\!\cdots\!151\)\(\medspace = 3^{10} \cdot 19^{5} \cdot 61^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.3404858678973.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 16T1294 |
Parity: | odd |
Determinant: | 1.1159.2t1.a.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.0.3404858678973.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + 9x^{6} - 13x^{5} - 27x^{4} + 36x^{3} + 121x^{2} - 117x + 27 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{3} + x + 28 \)
Roots:
$r_{ 1 }$ | $=$ | \( 21 a^{2} + 11 a + 18 + \left(2 a^{2} + 12 a + 27\right)\cdot 31 + \left(7 a^{2} + 14\right)\cdot 31^{2} + \left(11 a^{2} + 27 a + 8\right)\cdot 31^{3} + \left(18 a^{2} + 14 a + 28\right)\cdot 31^{4} + \left(6 a^{2} + 24 a + 12\right)\cdot 31^{5} + \left(24 a^{2} + 19 a + 13\right)\cdot 31^{6} + \left(25 a^{2} + 9 a + 17\right)\cdot 31^{7} + \left(11 a^{2} + 25 a + 29\right)\cdot 31^{8} + \left(30 a^{2} + 28 a + 17\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 17 a^{2} + 10 a + 3 + \left(23 a^{2} + 7 a + 15\right)\cdot 31 + \left(5 a^{2} + 18 a + 3\right)\cdot 31^{2} + \left(10 a^{2} + 18 a + 4\right)\cdot 31^{3} + \left(30 a^{2} + 15 a + 9\right)\cdot 31^{4} + \left(14 a^{2} + 13 a + 10\right)\cdot 31^{5} + \left(17 a + 26\right)\cdot 31^{6} + \left(10 a^{2} + 17 a + 4\right)\cdot 31^{7} + \left(6 a^{2} + 8 a + 28\right)\cdot 31^{8} + \left(11 a^{2} + 3 a + 9\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 18 a^{2} + 9 a + 16 + \left(7 a^{2} + 20 a + 20\right)\cdot 31 + \left(28 a^{2} + 24 a + 18\right)\cdot 31^{2} + \left(26 a^{2} + 14 a + 8\right)\cdot 31^{3} + \left(17 a + 6\right)\cdot 31^{4} + \left(14 a^{2} + 6 a + 28\right)\cdot 31^{5} + \left(7 a^{2} + 19 a + 22\right)\cdot 31^{6} + \left(25 a + 10\right)\cdot 31^{7} + \left(12 a^{2} + 29 a + 19\right)\cdot 31^{8} + \left(10 a^{2} + 12 a + 4\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 8 + 18\cdot 31 + 13\cdot 31^{2} + 10\cdot 31^{3} + 8\cdot 31^{4} + 8\cdot 31^{5} + 14\cdot 31^{6} + 4\cdot 31^{7} + 9\cdot 31^{8} + 9\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 9 a^{2} + 23 a + 8 + \left(2 a^{2} + 25 a + 11\right)\cdot 31 + \left(24 a^{2} + 16 a + 5\right)\cdot 31^{2} + \left(7 a^{2} + 16 a + 23\right)\cdot 31^{3} + \left(28 a + 9\right)\cdot 31^{4} + \left(25 a^{2} + 19 a + 27\right)\cdot 31^{5} + \left(8 a^{2} + 27 a\right)\cdot 31^{6} + \left(16 a^{2} + 18 a + 9\right)\cdot 31^{7} + \left(28 a^{2} + 2 a + 22\right)\cdot 31^{8} + \left(30 a^{2} + 18 a + 12\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 5 + 30\cdot 31 + 18\cdot 31^{2} + 25\cdot 31^{3} + 7\cdot 31^{4} + 27\cdot 31^{5} + 8\cdot 31^{6} + 9\cdot 31^{8} + 21\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 23 a^{2} + 11 a + 9 + \left(20 a^{2} + 29 a + 29\right)\cdot 31 + \left(26 a^{2} + 5 a + 27\right)\cdot 31^{2} + \left(23 a^{2} + 20 a + 16\right)\cdot 31^{3} + \left(11 a^{2} + 29 a + 13\right)\cdot 31^{4} + \left(10 a^{2} + 30 a + 15\right)\cdot 31^{5} + \left(30 a^{2} + 22 a + 17\right)\cdot 31^{6} + \left(4 a^{2} + 26 a + 3\right)\cdot 31^{7} + \left(7 a^{2} + 6 a + 16\right)\cdot 31^{8} + \left(21 a^{2} + 20 a + 1\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 5 a^{2} + 29 a + 26 + \left(5 a^{2} + 28 a + 2\right)\cdot 31 + \left(a^{2} + 26 a + 21\right)\cdot 31^{2} + \left(13 a^{2} + 26 a + 26\right)\cdot 31^{3} + \left(17 a + 9\right)\cdot 31^{4} + \left(22 a^{2} + 28 a + 25\right)\cdot 31^{5} + \left(21 a^{2} + 16 a + 19\right)\cdot 31^{6} + \left(4 a^{2} + 25 a + 11\right)\cdot 31^{7} + \left(27 a^{2} + 19 a + 21\right)\cdot 31^{8} + \left(19 a^{2} + 9 a + 15\right)\cdot 31^{9} +O(31^{10})\) |
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 |
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(1,4)(3,7)$ | $-3$ |
$9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $1$ |
$12$ | $2$ | $(2,5)$ | $3$ |
$24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $3$ |
$36$ | $2$ | $(1,3)(2,5)$ | $1$ |
$36$ | $2$ | $(1,4)(2,5)(3,7)$ | $-1$ |
$16$ | $3$ | $(2,6,8)$ | $0$ |
$64$ | $3$ | $(2,6,8)(3,4,7)$ | $0$ |
$12$ | $4$ | $(1,3,4,7)$ | $-3$ |
$36$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $1$ |
$36$ | $4$ | $(1,4)(2,5,6,8)(3,7)$ | $1$ |
$72$ | $4$ | $(1,6,4,2)(3,8,7,5)$ | $-1$ |
$72$ | $4$ | $(1,3,4,7)(2,5)$ | $-1$ |
$144$ | $4$ | $(1,2,3,5)(4,6)(7,8)$ | $1$ |
$48$ | $6$ | $(1,4)(2,8,6)(3,7)$ | $0$ |
$96$ | $6$ | $(2,5)(3,7,4)$ | $0$ |
$192$ | $6$ | $(1,5)(2,3,6,4,8,7)$ | $0$ |
$144$ | $8$ | $(1,5,3,6,4,8,7,2)$ | $-1$ |
$96$ | $12$ | $(1,3,4,7)(2,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.