Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(7953250761\)\(\medspace = 3^{10} \cdot 367^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.3.36035099127.1 |
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 9.3.36035099127.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 2x^{8} + x^{7} - 2x^{6} + 5x^{5} + 5x^{4} - 16x^{3} + 7x^{2} + 4x - 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: \( x^{3} + 7x + 59 \)
Roots:
$r_{ 1 }$ | $=$ | \( 7 a^{2} + 7 a + 21 + \left(51 a^{2} + 26 a + 22\right)\cdot 61 + \left(43 a^{2} + 31 a + 23\right)\cdot 61^{2} + \left(12 a^{2} + 21 a + 43\right)\cdot 61^{3} + \left(5 a^{2} + 48 a + 55\right)\cdot 61^{4} + \left(17 a^{2} + 22 a + 55\right)\cdot 61^{5} + \left(36 a^{2} + 45 a + 50\right)\cdot 61^{6} + \left(23 a^{2} + 46 a + 20\right)\cdot 61^{7} + \left(51 a^{2} + 44 a + 28\right)\cdot 61^{8} + \left(45 a^{2} + 47 a + 26\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 2 }$ | $=$ | \( 23 a^{2} + 59 a + 50 + \left(5 a^{2} + 14 a + 38\right)\cdot 61 + \left(60 a^{2} + 17 a + 28\right)\cdot 61^{2} + \left(12 a^{2} + 13 a + 9\right)\cdot 61^{3} + \left(21 a^{2} + 46 a + 48\right)\cdot 61^{4} + \left(56 a^{2} + 10 a + 31\right)\cdot 61^{5} + \left(38 a^{2} + 24 a + 11\right)\cdot 61^{6} + \left(33 a^{2} + 46 a + 27\right)\cdot 61^{7} + \left(18 a^{2} + 40 a + 35\right)\cdot 61^{8} + \left(13 a^{2} + 35 a + 44\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 3 }$ | $=$ | \( 51 a^{2} + 16 a + 18 + \left(38 a^{2} + 15 a + 32\right)\cdot 61 + \left(23 a^{2} + 36 a + 41\right)\cdot 61^{2} + \left(13 a^{2} + a + 31\right)\cdot 61^{3} + \left(21 a^{2} + 15 a + 7\right)\cdot 61^{4} + \left(55 a + 55\right)\cdot 61^{5} + \left(33 a^{2} + 49 a + 44\right)\cdot 61^{6} + \left(34 a^{2} + 52 a + 51\right)\cdot 61^{7} + \left(46 a^{2} + 31 a + 23\right)\cdot 61^{8} + \left(29 a^{2} + 51 a + 60\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 4 }$ | $=$ | \( 24 a^{2} + 9 a + 19 + \left(50 a^{2} + 5 a + 51\right)\cdot 61 + \left(28 a^{2} + 43 a + 38\right)\cdot 61^{2} + \left(23 a^{2} + 16 a + 54\right)\cdot 61^{3} + \left(48 a^{2} + 49 a + 41\right)\cdot 61^{4} + \left(45 a^{2} + 26 a + 41\right)\cdot 61^{5} + \left(46 a^{2} + 7 a + 18\right)\cdot 61^{6} + \left(31 a^{2} + 16 a + 1\right)\cdot 61^{7} + \left(46 a^{2} + 9 a + 12\right)\cdot 61^{8} + \left(53 a^{2} + 15 a + 49\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 5 }$ | $=$ | \( 47 a^{2} + 31 a + 45 + \left(25 a^{2} + 10 a + 17\right)\cdot 61 + \left(21 a^{2} + 34 a + 4\right)\cdot 61^{2} + \left(42 a^{2} + 21 a + 41\right)\cdot 61^{3} + \left(23 a^{2} + 3 a + 48\right)\cdot 61^{4} + \left(55 a^{2} + 30 a + 45\right)\cdot 61^{5} + \left(59 a^{2} + 55 a + 59\right)\cdot 61^{6} + \left(39 a^{2} + 7 a + 59\right)\cdot 61^{7} + \left(50 a^{2} + 17 a + 10\right)\cdot 61^{8} + \left(52 a^{2} + 43 a + 4\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 6 }$ | $=$ | \( 40 a^{2} + 39 a + 53 + \left(42 a^{2} + 32 a + 43\right)\cdot 61 + \left(44 a^{2} + 21 a + 47\right)\cdot 61^{2} + \left(21 a^{2} + 12 a + 44\right)\cdot 61^{3} + \left(10 a^{2} + 38 a + 18\right)\cdot 61^{4} + \left(8 a^{2} + 13 a + 14\right)\cdot 61^{5} + \left(14 a^{2} + 31 a + 49\right)\cdot 61^{6} + \left(24 a^{2} + 25 a + 23\right)\cdot 61^{7} + \left(50 a^{2} + 28 a + 3\right)\cdot 61^{8} + \left(25 a^{2} + 27 a + 55\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 7 }$ | $=$ | \( 48 a^{2} + 47 a + 4 + \left(16 a^{2} + 30 a + 31\right)\cdot 61 + \left(38 a^{2} + 7 a + 28\right)\cdot 61^{2} + \left(34 a^{2} + 46 a + 49\right)\cdot 61^{3} + \left(18 a^{2} + 60 a + 35\right)\cdot 61^{4} + \left(4 a^{2} + 55 a + 12\right)\cdot 61^{5} + \left(50 a^{2} + 47 a + 43\right)\cdot 61^{6} + \left(53 a^{2} + 22 a + 19\right)\cdot 61^{7} + \left(56 a^{2} + 49 a + 31\right)\cdot 61^{8} + \left(17 a^{2} + 34 a + 25\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 8 }$ | $=$ | \( 14 a^{2} + 15 a + 13 + \left(28 a^{2} + 2 a + 17\right)\cdot 61 + \left(33 a^{2} + 8 a + 56\right)\cdot 61^{2} + \left(26 a^{2} + 27 a + 46\right)\cdot 61^{3} + \left(45 a^{2} + 35 a + 19\right)\cdot 61^{4} + \left(35 a^{2} + 24 a + 41\right)\cdot 61^{5} + \left(10 a^{2} + 45 a + 32\right)\cdot 61^{6} + \left(13 a^{2} + 49 a + 53\right)\cdot 61^{7} + \left(20 a^{2} + 48 a + 4\right)\cdot 61^{8} + \left(50 a^{2} + 46 a + 27\right)\cdot 61^{9} +O(61^{10})\) |
$r_{ 9 }$ | $=$ | \( 51 a^{2} + 21 a + 23 + \left(45 a^{2} + 45 a + 50\right)\cdot 61 + \left(10 a^{2} + 44 a + 35\right)\cdot 61^{2} + \left(56 a^{2} + 22 a + 44\right)\cdot 61^{3} + \left(49 a^{2} + 8 a + 28\right)\cdot 61^{4} + \left(20 a^{2} + 4 a + 6\right)\cdot 61^{5} + \left(15 a^{2} + 59 a + 55\right)\cdot 61^{6} + \left(50 a^{2} + 36 a + 46\right)\cdot 61^{7} + \left(24 a^{2} + 34 a + 32\right)\cdot 61^{8} + \left(15 a^{2} + 2 a + 12\right)\cdot 61^{9} +O(61^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(1,4)$ | $0$ |
$18$ | $2$ | $(1,2)(4,8)(7,9)$ | $4$ |
$27$ | $2$ | $(1,4)(2,8)(3,5)$ | $0$ |
$27$ | $2$ | $(1,4)(2,8)$ | $0$ |
$54$ | $2$ | $(1,4)(2,3)(5,8)(6,9)$ | $0$ |
$6$ | $3$ | $(3,5,6)$ | $-4$ |
$8$ | $3$ | $(1,4,7)(2,8,9)(3,5,6)$ | $-1$ |
$12$ | $3$ | $(2,8,9)(3,5,6)$ | $2$ |
$72$ | $3$ | $(1,2,3)(4,8,5)(6,7,9)$ | $2$ |
$54$ | $4$ | $(1,8,4,2)(7,9)$ | $0$ |
$162$ | $4$ | $(1,5,4,3)(6,7)(8,9)$ | $0$ |
$36$ | $6$ | $(1,2)(3,5,6)(4,8)(7,9)$ | $-2$ |
$36$ | $6$ | $(1,3,4,5,7,6)$ | $-2$ |
$36$ | $6$ | $(1,4)(3,5,6)$ | $0$ |
$36$ | $6$ | $(1,4)(2,8,9)(3,5,6)$ | $0$ |
$54$ | $6$ | $(1,4)(2,8)(3,6,5)$ | $0$ |
$72$ | $6$ | $(1,2,4,8,7,9)(3,5,6)$ | $1$ |
$108$ | $6$ | $(1,4)(2,3,8,5,9,6)$ | $0$ |
$216$ | $6$ | $(1,8,5,4,2,3)(6,7,9)$ | $0$ |
$144$ | $9$ | $(1,2,3,4,8,5,7,9,6)$ | $-1$ |
$108$ | $12$ | $(1,8,4,2)(3,5,6)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.