Basic invariants
Dimension: | $9$ |
Group: | $S_4\wr C_2$ |
Conductor: | \(32454737997240833\)\(\medspace = 37^{6} \cdot 233^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.4.109049934277.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T272 |
Parity: | even |
Determinant: | 1.233.2t1.a.a |
Projective image: | $S_4\wr C_2$ |
Projective stem field: | Galois closure of 8.4.109049934277.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + x^{6} - 7x^{5} + 20x^{4} - 13x^{3} - 46x^{2} + 71x - 16 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{2} + 82x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 30 a + 70 + \left(23 a + 4\right)\cdot 89 + 53 a\cdot 89^{2} + \left(34 a + 27\right)\cdot 89^{3} + \left(71 a + 74\right)\cdot 89^{4} + \left(13 a + 53\right)\cdot 89^{5} + \left(31 a + 87\right)\cdot 89^{6} + \left(16 a + 54\right)\cdot 89^{7} + \left(77 a + 74\right)\cdot 89^{8} + \left(72 a + 54\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 2 }$ | $=$ | \( 30 a + 83 + \left(49 a + 16\right)\cdot 89 + \left(28 a + 86\right)\cdot 89^{2} + \left(a + 33\right)\cdot 89^{3} + \left(74 a + 16\right)\cdot 89^{4} + \left(55 a + 50\right)\cdot 89^{5} + \left(57 a + 40\right)\cdot 89^{6} + \left(51 a + 20\right)\cdot 89^{7} + \left(65 a + 75\right)\cdot 89^{8} + \left(75 a + 32\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 3 }$ | $=$ | \( 59 a + 26 + \left(39 a + 65\right)\cdot 89 + \left(60 a + 58\right)\cdot 89^{2} + \left(87 a + 14\right)\cdot 89^{3} + \left(14 a + 88\right)\cdot 89^{4} + \left(33 a + 10\right)\cdot 89^{5} + \left(31 a + 32\right)\cdot 89^{6} + \left(37 a + 57\right)\cdot 89^{7} + \left(23 a + 37\right)\cdot 89^{8} + \left(13 a + 52\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 4 }$ | $=$ | \( 88 + 13\cdot 89 + 57\cdot 89^{2} + 82\cdot 89^{3} + 66\cdot 89^{4} + 89^{5} + 82\cdot 89^{6} + 26\cdot 89^{7} + 37\cdot 89^{8} + 31\cdot 89^{9} +O(89^{10})\) |
$r_{ 5 }$ | $=$ | \( 8 + 21\cdot 89 + 38\cdot 89^{2} + 30\cdot 89^{3} + 31\cdot 89^{4} + 43\cdot 89^{5} + 72\cdot 89^{6} + 46\cdot 89^{7} + 89^{8} + 49\cdot 89^{9} +O(89^{10})\) |
$r_{ 6 }$ | $=$ | \( 59 a + 13 + \left(65 a + 49\right)\cdot 89 + \left(35 a + 82\right)\cdot 89^{2} + \left(54 a + 37\right)\cdot 89^{3} + \left(17 a + 5\right)\cdot 89^{4} + \left(75 a + 79\right)\cdot 89^{5} + \left(57 a + 24\right)\cdot 89^{6} + \left(72 a + 49\right)\cdot 89^{7} + \left(11 a + 64\right)\cdot 89^{8} + \left(16 a + 42\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 7 }$ | $=$ | \( 38 a + 80 + \left(53 a + 57\right)\cdot 89 + \left(60 a + 53\right)\cdot 89^{2} + \left(71 a + 66\right)\cdot 89^{3} + \left(82 a + 49\right)\cdot 89^{4} + \left(67 a + 84\right)\cdot 89^{5} + \left(3 a + 28\right)\cdot 89^{6} + \left(43 a + 79\right)\cdot 89^{7} + \left(23 a + 60\right)\cdot 89^{8} + \left(60 a + 69\right)\cdot 89^{9} +O(89^{10})\) |
$r_{ 8 }$ | $=$ | \( 51 a + 79 + \left(35 a + 37\right)\cdot 89 + \left(28 a + 68\right)\cdot 89^{2} + \left(17 a + 62\right)\cdot 89^{3} + \left(6 a + 23\right)\cdot 89^{4} + \left(21 a + 32\right)\cdot 89^{5} + \left(85 a + 76\right)\cdot 89^{6} + \left(45 a + 20\right)\cdot 89^{7} + \left(65 a + 4\right)\cdot 89^{8} + \left(28 a + 23\right)\cdot 89^{9} +O(89^{10})\) |
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 |
$1$ | $1$ | $()$ | $9$ |
$6$ | $2$ | $(2,7)(3,8)$ | $-3$ |
$9$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $1$ |
$12$ | $2$ | $(1,4)$ | $-3$ |
$24$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $3$ |
$36$ | $2$ | $(1,4)(2,3)$ | $1$ |
$36$ | $2$ | $(1,4)(2,7)(3,8)$ | $1$ |
$16$ | $3$ | $(1,5,6)$ | $0$ |
$64$ | $3$ | $(1,5,6)(3,7,8)$ | $0$ |
$12$ | $4$ | $(2,3,7,8)$ | $3$ |
$36$ | $4$ | $(1,4,5,6)(2,3,7,8)$ | $1$ |
$36$ | $4$ | $(1,4,5,6)(2,7)(3,8)$ | $-1$ |
$72$ | $4$ | $(1,2,5,7)(3,6,8,4)$ | $-1$ |
$72$ | $4$ | $(1,4)(2,3,7,8)$ | $-1$ |
$144$ | $4$ | $(1,3,4,2)(5,7)(6,8)$ | $-1$ |
$48$ | $6$ | $(1,6,5)(2,7)(3,8)$ | $0$ |
$96$ | $6$ | $(1,4)(3,8,7)$ | $0$ |
$192$ | $6$ | $(1,3,5,7,6,8)(2,4)$ | $0$ |
$144$ | $8$ | $(1,2,4,3,5,7,6,8)$ | $1$ |
$96$ | $12$ | $(1,5,6)(2,3,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.