Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(119738145024\)\(\medspace = 2^{8} \cdot 3^{10} \cdot 89^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.131564134656.1 |
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 9.1.131564134656.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} + 2x^{7} - 8x^{6} + 10x^{4} + 20x^{3} - 12x^{2} - 28x - 10 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{3} + 6x + 65 \)
Roots:
$r_{ 1 }$ | $=$ | \( 5 a^{2} + 60 a + 39 + \left(59 a^{2} + 59 a + 30\right)\cdot 67 + \left(31 a^{2} + 44 a + 4\right)\cdot 67^{2} + \left(58 a^{2} + 38 a + 46\right)\cdot 67^{3} + \left(66 a^{2} + 26 a + 8\right)\cdot 67^{4} + \left(12 a^{2} + 20 a + 44\right)\cdot 67^{5} + \left(51 a^{2} + 12 a + 65\right)\cdot 67^{6} + \left(10 a^{2} + 34 a + 2\right)\cdot 67^{7} + \left(63 a^{2} + 54 a + 48\right)\cdot 67^{8} + \left(10 a^{2} + 60 a + 50\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 2 }$ | $=$ | \( 64 a^{2} + 44 a + 56 + \left(46 a^{2} + 8 a + 57\right)\cdot 67 + \left(60 a^{2} + 25 a + 54\right)\cdot 67^{2} + \left(51 a^{2} + 30 a + 1\right)\cdot 67^{3} + \left(23 a^{2} + 46 a + 60\right)\cdot 67^{4} + \left(15 a^{2} + 64 a + 7\right)\cdot 67^{5} + \left(18 a^{2} + 43 a + 13\right)\cdot 67^{6} + \left(32 a^{2} + 5 a + 44\right)\cdot 67^{7} + \left(63 a^{2} + 47 a + 26\right)\cdot 67^{8} + \left(42 a + 43\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 3 }$ | $=$ | \( 34 a^{2} + 42 a + 21 + \left(65 a^{2} + 66 a + 56\right)\cdot 67 + \left(25 a^{2} + 42 a + 47\right)\cdot 67^{2} + \left(12 a^{2} + 51 a + 62\right)\cdot 67^{3} + \left(30 a^{2} + 44 a + 62\right)\cdot 67^{4} + \left(16 a^{2} + 47 a + 57\right)\cdot 67^{5} + \left(20 a^{2} + 21 a + 8\right)\cdot 67^{6} + \left(43 a^{2} + 36 a + 66\right)\cdot 67^{7} + \left(8 a^{2} + 15 a + 30\right)\cdot 67^{8} + \left(12 a^{2} + 45 a + 55\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 4 }$ | $=$ | \( 28 a^{2} + 32 a + 64 + \left(9 a^{2} + 7 a + 32\right)\cdot 67 + \left(9 a^{2} + 46 a + 47\right)\cdot 67^{2} + \left(63 a^{2} + 43 a + 64\right)\cdot 67^{3} + \left(36 a^{2} + 62 a + 22\right)\cdot 67^{4} + \left(37 a^{2} + 65 a + 8\right)\cdot 67^{5} + \left(62 a^{2} + 32 a + 44\right)\cdot 67^{6} + \left(12 a^{2} + 63 a + 11\right)\cdot 67^{7} + \left(62 a^{2} + 63 a + 44\right)\cdot 67^{8} + \left(43 a^{2} + 27 a + 48\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 5 }$ | $=$ | \( 36 a^{2} + 48 a + 57 + \left(21 a^{2} + 58 a + 19\right)\cdot 67 + \left(47 a^{2} + 65 a + 31\right)\cdot 67^{2} + \left(2 a^{2} + 51 a + 2\right)\cdot 67^{3} + \left(13 a^{2} + 42 a + 11\right)\cdot 67^{4} + \left(35 a^{2} + 21 a + 1\right)\cdot 67^{5} + \left(28 a^{2} + a + 45\right)\cdot 67^{6} + \left(58 a^{2} + 25 a + 23\right)\cdot 67^{7} + \left(61 a^{2} + 4 a + 9\right)\cdot 67^{8} + \left(53 a^{2} + 46 a + 35\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 6 }$ | $=$ | \( 43 a + 1 + \left(24 a^{2} + 56 a + 33\right)\cdot 67 + \left(16 a^{2} + 28 a + 11\right)\cdot 67^{2} + \left(7 a^{2} + 43 a + 24\right)\cdot 67^{3} + \left(22 a^{2} + 31 a + 53\right)\cdot 67^{4} + \left(12 a^{2} + 40 a + 62\right)\cdot 67^{5} + \left(60 a^{2} + 2 a + 46\right)\cdot 67^{6} + \left(9 a^{2} + 31 a + 21\right)\cdot 67^{7} + \left(58 a^{2} + 56 a + 5\right)\cdot 67^{8} + \left(5 a^{2} + 21 a + 63\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 7 }$ | $=$ | \( 36 a^{2} + 55 a + 57 + \left(61 a^{2} + 57 a + 45\right)\cdot 67 + \left(7 a + 46\right)\cdot 67^{2} + \left(63 a^{2} + 30 a + 42\right)\cdot 67^{3} + \left(8 a^{2} + 15 a + 61\right)\cdot 67^{4} + \left(57 a^{2} + 39 a + 21\right)\cdot 67^{5} + \left(15 a^{2} + 13 a + 61\right)\cdot 67^{6} + \left(29 a^{2} + 40 a + 40\right)\cdot 67^{7} + \left(59 a^{2} + 39 a + 66\right)\cdot 67^{8} + \left(29 a^{2} + 36 a + 5\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 8 }$ | $=$ | \( 62 a^{2} + 31 a + 27 + \left(50 a^{2} + 17 a + 3\right)\cdot 67 + \left(18 a^{2} + 60 a + 51\right)\cdot 67^{2} + \left(a^{2} + 51 a + 63\right)\cdot 67^{3} + \left(45 a^{2} + 8 a + 4\right)\cdot 67^{4} + \left(41 a^{2} + 6 a + 27\right)\cdot 67^{5} + \left(22 a^{2} + 52 a + 21\right)\cdot 67^{6} + \left(46 a^{2} + a + 42\right)\cdot 67^{7} + \left(12 a^{2} + 23 a + 13\right)\cdot 67^{8} + \left(50 a^{2} + 51 a + 20\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 9 }$ | $=$ | \( 3 a^{2} + 47 a + 13 + \left(63 a^{2} + a + 55\right)\cdot 67 + \left(56 a^{2} + 13 a + 39\right)\cdot 67^{2} + \left(7 a^{2} + 60 a + 26\right)\cdot 67^{3} + \left(21 a^{2} + 55 a + 49\right)\cdot 67^{4} + \left(39 a^{2} + 28 a + 36\right)\cdot 67^{5} + \left(55 a^{2} + 20 a + 28\right)\cdot 67^{6} + \left(24 a^{2} + 30 a + 14\right)\cdot 67^{7} + \left(12 a^{2} + 30 a + 23\right)\cdot 67^{8} + \left(60 a^{2} + 2 a + 12\right)\cdot 67^{9} +O(67^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(4,7)$ | $0$ |
$18$ | $2$ | $(1,4)(6,7)(8,9)$ | $4$ |
$27$ | $2$ | $(1,6)(4,7)$ | $0$ |
$27$ | $2$ | $(1,6)(2,3)(4,7)$ | $0$ |
$54$ | $2$ | $(1,6)(2,4)(3,7)(5,9)$ | $0$ |
$6$ | $3$ | $(2,3,5)$ | $-4$ |
$8$ | $3$ | $(1,6,8)(2,3,5)(4,7,9)$ | $-1$ |
$12$ | $3$ | $(2,3,5)(4,7,9)$ | $2$ |
$72$ | $3$ | $(1,2,4)(3,7,6)(5,9,8)$ | $2$ |
$54$ | $4$ | $(1,7,6,4)(8,9)$ | $0$ |
$162$ | $4$ | $(1,6)(2,4,3,7)(5,9)$ | $0$ |
$36$ | $6$ | $(1,4)(2,3,5)(6,7)(8,9)$ | $-2$ |
$36$ | $6$ | $(2,7,3,9,5,4)$ | $-2$ |
$36$ | $6$ | $(2,3,5)(4,7)$ | $0$ |
$36$ | $6$ | $(1,6,8)(2,3,5)(4,7)$ | $0$ |
$54$ | $6$ | $(1,6)(2,5,3)(4,7)$ | $0$ |
$72$ | $6$ | $(1,9,8,7,6,4)(2,3,5)$ | $1$ |
$108$ | $6$ | $(1,6)(2,7,3,9,5,4)$ | $0$ |
$216$ | $6$ | $(1,2,4,6,3,7)(5,9,8)$ | $0$ |
$144$ | $9$ | $(1,2,7,6,3,9,8,5,4)$ | $-1$ |
$108$ | $12$ | $(1,7,6,4)(2,3,5)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.