Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(1546691584\)\(\medspace = 2^{10} \cdot 1229^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.3.118805247296.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T213 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.3.118805247296.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 2x^{8} - x^{7} - x^{6} + 19x^{5} - 41x^{4} + 50x^{3} - 35x^{2} + 4x + 8 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: \( x^{3} + 3x + 86 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 12 a^{2} + 20 a + 68 + \left(11 a^{2} + 42 a\right)\cdot 89 + \left(71 a^{2} + 46 a + 7\right)\cdot 89^{2} + \left(14 a^{2} + 66 a + 5\right)\cdot 89^{3} + \left(60 a^{2} + 21 a + 77\right)\cdot 89^{4} + \left(24 a^{2} + 64 a + 38\right)\cdot 89^{5} + \left(5 a^{2} + 42 a + 74\right)\cdot 89^{6} + \left(7 a^{2} + 60 a + 28\right)\cdot 89^{7} + \left(82 a^{2} + 46 a + 20\right)\cdot 89^{8} + \left(49 a^{2} + 17 a\right)\cdot 89^{9} +O(89^{10})\)
$r_{ 2 }$ |
$=$ |
\( 68 a^{2} + 31 a + 2 + \left(13 a^{2} + 82 a + 6\right)\cdot 89 + \left(39 a^{2} + 57 a + 32\right)\cdot 89^{2} + \left(47 a^{2} + 29 a + 70\right)\cdot 89^{3} + \left(3 a^{2} + 45 a + 52\right)\cdot 89^{4} + \left(50 a^{2} + 64 a\right)\cdot 89^{5} + \left(21 a^{2} + 15 a + 18\right)\cdot 89^{6} + \left(72 a^{2} + 46 a + 70\right)\cdot 89^{7} + \left(74 a^{2} + 71 a + 5\right)\cdot 89^{8} + \left(62 a^{2} + 60 a + 26\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 23 a^{2} + 29 a + 8 + \left(56 a^{2} + 6 a + 61\right)\cdot 89 + \left(64 a^{2} + 5 a + 59\right)\cdot 89^{2} + \left(78 a^{2} + 79 a + 70\right)\cdot 89^{3} + \left(83 a + 28\right)\cdot 89^{4} + \left(5 a^{2} + 22 a + 84\right)\cdot 89^{5} + \left(69 a^{2} + 31 a + 62\right)\cdot 89^{6} + \left(52 a^{2} + 38 a + 57\right)\cdot 89^{7} + \left(12 a^{2} + 41 a + 19\right)\cdot 89^{8} + \left(50 a^{2} + 26 a + 42\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 38 a^{2} + 62 a + 41 + \left(60 a^{2} + 35 a + 45\right)\cdot 89 + \left(78 a^{2} + 70 a + 65\right)\cdot 89^{2} + \left(39 a^{2} + 45 a + 42\right)\cdot 89^{3} + \left(39 a^{2} + 29 a + 65\right)\cdot 89^{4} + \left(20 a^{2} + 4 a + 6\right)\cdot 89^{5} + \left(26 a^{2} + 80 a + 34\right)\cdot 89^{6} + \left(54 a^{2} + 9 a + 82\right)\cdot 89^{7} + \left(10 a^{2} + 40 a + 51\right)\cdot 89^{8} + \left(75 a^{2} + 61 a + 70\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 9 a^{2} + 38 a + 62 + \left(64 a^{2} + 53 a + 17\right)\cdot 89 + \left(67 a^{2} + 73 a\right)\cdot 89^{2} + \left(26 a^{2} + 81 a + 29\right)\cdot 89^{3} + \left(25 a^{2} + 21 a + 7\right)\cdot 89^{4} + \left(14 a^{2} + 49 a + 18\right)\cdot 89^{5} + \left(62 a^{2} + 30 a + 10\right)\cdot 89^{6} + \left(9 a^{2} + 71 a + 34\right)\cdot 89^{7} + \left(21 a^{2} + 59 a + 76\right)\cdot 89^{8} + \left(65 a^{2} + 10 a + 30\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 63 a^{2} + 17 a + 2 + \left(28 a^{2} + 84 a + 71\right)\cdot 89 + \left(75 a^{2} + 20 a + 58\right)\cdot 89^{2} + \left(59 a^{2} + 54 a + 82\right)\cdot 89^{3} + \left(37 a^{2} + 73 a + 61\right)\cdot 89^{4} + \left(84 a^{2} + 51 a + 45\right)\cdot 89^{5} + \left(27 a^{2} + 20 a + 37\right)\cdot 89^{6} + \left(31 a + 63\right)\cdot 89^{7} + \left(33 a^{2} + 55 a + 7\right)\cdot 89^{8} + \left(5 a^{2} + 57 a + 20\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 14 a^{2} + 17 a + 79 + \left(59 a^{2} + 82 a + 66\right)\cdot 89 + \left(3 a^{2} + 19 a + 26\right)\cdot 89^{2} + \left(51 a^{2} + 16 a + 15\right)\cdot 89^{3} + \left(15 a^{2} + 84 a + 58\right)\cdot 89^{4} + \left(47 a^{2} + 82 a + 79\right)\cdot 89^{5} + \left(82 a^{2} + 63 a\right)\cdot 89^{6} + \left(14 a^{2} + 38 a + 71\right)\cdot 89^{7} + \left(87 a^{2} + 35 a + 79\right)\cdot 89^{8} + \left(25 a^{2} + 71 a + 82\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 52 a^{2} + 43 a + 66 + \left(62 a^{2} + 73\right)\cdot 89 + \left(20 a^{2} + 64 a + 60\right)\cdot 89^{2} + \left(48 a^{2} + 82 a + 9\right)\cdot 89^{3} + \left(72 a^{2} + 9 a + 83\right)\cdot 89^{4} + \left(36 a^{2} + 72 a + 58\right)\cdot 89^{5} + \left(26 a^{2} + 82 a + 66\right)\cdot 89^{6} + \left(21 a^{2} + 11 a + 83\right)\cdot 89^{7} + \left(78 a^{2} + 12 a + 61\right)\cdot 89^{8} + \left(12 a^{2} + 80 a + 56\right)\cdot 89^{9} +O(89^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 77 a^{2} + 10 a + 30 + \left(88 a^{2} + 58 a + 13\right)\cdot 89 + \left(23 a^{2} + 86 a + 45\right)\cdot 89^{2} + \left(78 a^{2} + 77 a + 30\right)\cdot 89^{3} + \left(11 a^{2} + 74 a + 10\right)\cdot 89^{4} + \left(73 a^{2} + 32 a + 23\right)\cdot 89^{5} + \left(34 a^{2} + 77 a + 51\right)\cdot 89^{6} + \left(34 a^{2} + 47 a + 42\right)\cdot 89^{7} + \left(45 a^{2} + 82 a + 32\right)\cdot 89^{8} + \left(8 a^{2} + 58 a + 26\right)\cdot 89^{9} +O(89^{10})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(1,4)$ | $0$ |
$18$ | $2$ | $(1,2)(3,4)(7,9)$ | $4$ |
$27$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
$27$ | $2$ | $(1,4)(2,3)$ | $0$ |
$54$ | $2$ | $(1,4)(2,5)(3,6)(8,9)$ | $0$ |
$6$ | $3$ | $(5,6,8)$ | $-4$ |
$8$ | $3$ | $(1,4,7)(2,3,9)(5,6,8)$ | $-1$ |
$12$ | $3$ | $(1,4,7)(5,6,8)$ | $2$ |
$72$ | $3$ | $(1,2,5)(3,6,4)(7,9,8)$ | $2$ |
$54$ | $4$ | $(1,3,4,2)(7,9)$ | $0$ |
$162$ | $4$ | $(1,6,4,5)(3,9)(7,8)$ | $0$ |
$36$ | $6$ | $(1,2)(3,4)(5,6,8)(7,9)$ | $-2$ |
$36$ | $6$ | $(1,5,4,6,7,8)$ | $-2$ |
$36$ | $6$ | $(1,4)(5,6,8)$ | $0$ |
$36$ | $6$ | $(1,4)(2,3,9)(5,6,8)$ | $0$ |
$54$ | $6$ | $(1,4)(2,3)(5,8,6)$ | $0$ |
$72$ | $6$ | $(1,2,4,3,7,9)(5,6,8)$ | $1$ |
$108$ | $6$ | $(1,4)(2,5,3,6,9,8)$ | $0$ |
$216$ | $6$ | $(1,3,6,4,2,5)(7,9,8)$ | $0$ |
$144$ | $9$ | $(1,2,5,4,3,6,7,9,8)$ | $-1$ |
$108$ | $12$ | $(1,3,4,2)(5,6,8)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.