Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(7575961600\)\(\medspace = 2^{20} \cdot 5^{2} \cdot 17^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.20123648000.1 |
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 9.1.20123648000.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 2x^{8} + 5x^{7} + 2x^{6} + 2x^{5} + 6x^{4} + 2x^{3} + 4x^{2} + 8x + 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{3} + 3x + 51 \)
Roots:
$r_{ 1 }$ | $=$ | \( 50 a^{2} + 21 a + 52 + \left(a^{2} + 6 a + 48\right)\cdot 53 + \left(46 a^{2} + 31 a + 25\right)\cdot 53^{2} + \left(5 a^{2} + 15 a + 26\right)\cdot 53^{3} + \left(10 a^{2} + 48 a + 48\right)\cdot 53^{4} + \left(20 a^{2} + 50 a + 30\right)\cdot 53^{5} + \left(9 a^{2} + 11 a + 50\right)\cdot 53^{6} + \left(8 a^{2} + 45 a + 5\right)\cdot 53^{7} + \left(9 a^{2} + 30 a + 12\right)\cdot 53^{8} + \left(8 a^{2} + 31 a + 2\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 2 }$ | $=$ | \( 35 a^{2} + 8 a + 22 + \left(44 a^{2} + 10 a + 28\right)\cdot 53 + \left(23 a^{2} + 39 a + 34\right)\cdot 53^{2} + \left(14 a^{2} + 4 a + 43\right)\cdot 53^{3} + \left(30 a^{2} + 35 a + 35\right)\cdot 53^{4} + \left(34 a^{2} + 32 a + 6\right)\cdot 53^{5} + \left(52 a^{2} + 31 a + 31\right)\cdot 53^{6} + \left(42 a^{2} + 52 a + 22\right)\cdot 53^{7} + \left(48 a^{2} + 41 a + 38\right)\cdot 53^{8} + \left(51 a^{2} + 7 a + 36\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 3 }$ | $=$ | \( 32 a^{2} + 30 a + 48 + \left(12 a^{2} + 52 a + 19\right)\cdot 53 + \left(11 a^{2} + 16 a + 18\right)\cdot 53^{2} + \left(42 a^{2} + 6 a + 45\right)\cdot 53^{3} + \left(30 a^{2} + 31 a + 47\right)\cdot 53^{4} + \left(31 a^{2} + 5 a + 47\right)\cdot 53^{5} + \left(2 a^{2} + 51 a + 27\right)\cdot 53^{6} + \left(37 a^{2} + 24 a + 49\right)\cdot 53^{7} + \left(28 a^{2} + 17 a + 22\right)\cdot 53^{8} + \left(9 a^{2} + 47 a + 22\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 4 }$ | $=$ | \( 32 a^{2} + 34 a + 48 + \left(47 a^{2} + 35 a + 36\right)\cdot 53 + \left(30 a^{2} + 21 a + 4\right)\cdot 53^{2} + \left(3 a^{2} + 40 a + 21\right)\cdot 53^{3} + \left(50 a^{2} + 2 a + 33\right)\cdot 53^{4} + \left(34 a + 39\right)\cdot 53^{5} + \left(26 a^{2} + 30 a + 21\right)\cdot 53^{6} + \left(28 a^{2} + 25 a + 32\right)\cdot 53^{7} + \left(4 a^{2} + 22 a + 27\right)\cdot 53^{8} + \left(12 a^{2} + 47 a + 27\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 5 }$ | $=$ | \( 13 a^{2} + 20 a + 20 + \left(13 a^{2} + 33 a + 4\right)\cdot 53 + \left(26 a^{2} + 17 a + 52\right)\cdot 53^{2} + \left(a^{2} + 37 a + 44\right)\cdot 53^{3} + \left(28 a^{2} + 36 a + 23\right)\cdot 53^{4} + \left(23 a^{2} + a + 36\right)\cdot 53^{5} + \left(32 a^{2} + 32 a + 45\right)\cdot 53^{6} + \left(5 a^{2} + 8 a + 10\right)\cdot 53^{7} + \left(44 a^{2} + 50 a + 5\right)\cdot 53^{8} + \left(36 a^{2} + 37 a + 49\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 6 }$ | $=$ | \( 42 a^{2} + 42 a + 15 + \left(45 a^{2} + 17 a + 33\right)\cdot 53 + \left(10 a^{2} + 14 a + 17\right)\cdot 53^{2} + \left(7 a^{2} + 6 a + 28\right)\cdot 53^{3} + \left(25 a^{2} + 19 a + 36\right)\cdot 53^{4} + \left(20 a^{2} + 13 a + 25\right)\cdot 53^{5} + \left(24 a^{2} + 24 a + 18\right)\cdot 53^{6} + \left(40 a^{2} + 2 a + 3\right)\cdot 53^{7} + \left(19 a^{2} + 13 a + 5\right)\cdot 53^{8} + \left(31 a^{2} + 11 a + 13\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 7 }$ | $=$ | \( 11 a^{2} + 46 a + 16 + \left(3 a^{2} + 9 a + 37\right)\cdot 53 + \left(40 a^{2} + 13 a + 26\right)\cdot 53^{2} + \left(20 a^{2} + 39 a + 30\right)\cdot 53^{3} + \left(27 a^{2} + 22\right)\cdot 53^{4} + \left(4 a^{2} + 30 a + 51\right)\cdot 53^{5} + \left(21 a^{2} + 2 a + 22\right)\cdot 53^{6} + \left(34 a^{2} + 31 a + 15\right)\cdot 53^{7} + \left(43 a^{2} + 45 a + 4\right)\cdot 53^{8} + \left(23 a^{2} + 47 a + 23\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 8 }$ | $=$ | \( 29 a^{2} + 40 a + 52 + \left(36 a^{2} + 9 a + 50\right)\cdot 53 + \left(39 a^{2} + 22 a + 25\right)\cdot 53^{2} + \left(30 a^{2} + 29 a + 50\right)\cdot 53^{3} + \left(50 a^{2} + 15 a + 15\right)\cdot 53^{4} + \left(24 a^{2} + 21 a + 39\right)\cdot 53^{5} + \left(52 a^{2} + 18 a + 32\right)\cdot 53^{6} + \left(12 a^{2} + 13 a + 25\right)\cdot 53^{7} + \left(18 a^{2} + 10 a + 6\right)\cdot 53^{8} + \left(45 a^{2} + 20 a + 13\right)\cdot 53^{9} +O(53^{10})\) |
$r_{ 9 }$ | $=$ | \( 21 a^{2} + 24 a + 47 + \left(6 a^{2} + 36 a + 4\right)\cdot 53 + \left(36 a^{2} + 35 a + 6\right)\cdot 53^{2} + \left(32 a^{2} + 32 a + 27\right)\cdot 53^{3} + \left(12 a^{2} + 22 a\right)\cdot 53^{4} + \left(51 a^{2} + 22 a + 40\right)\cdot 53^{5} + \left(43 a^{2} + 9 a + 13\right)\cdot 53^{6} + \left(a^{2} + 8 a + 46\right)\cdot 53^{7} + \left(48 a^{2} + 33 a + 36\right)\cdot 53^{8} + \left(45 a^{2} + 13 a + 24\right)\cdot 53^{9} +O(53^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(3,7)$ | $0$ |
$18$ | $2$ | $(1,3)(4,7)(8,9)$ | $4$ |
$27$ | $2$ | $(1,4)(3,7)$ | $0$ |
$27$ | $2$ | $(1,4)(2,5)(3,7)$ | $0$ |
$54$ | $2$ | $(1,4)(2,3)(5,7)(6,9)$ | $0$ |
$6$ | $3$ | $(2,5,6)$ | $-4$ |
$8$ | $3$ | $(1,4,8)(2,5,6)(3,7,9)$ | $-1$ |
$12$ | $3$ | $(2,5,6)(3,7,9)$ | $2$ |
$72$ | $3$ | $(1,2,3)(4,5,7)(6,9,8)$ | $2$ |
$54$ | $4$ | $(1,7,4,3)(8,9)$ | $0$ |
$162$ | $4$ | $(1,4)(2,3,5,7)(6,9)$ | $0$ |
$36$ | $6$ | $(1,3)(2,5,6)(4,7)(8,9)$ | $-2$ |
$36$ | $6$ | $(2,7,5,9,6,3)$ | $-2$ |
$36$ | $6$ | $(2,5,6)(3,7)$ | $0$ |
$36$ | $6$ | $(1,4,8)(2,5,6)(3,7)$ | $0$ |
$54$ | $6$ | $(1,4)(2,6,5)(3,7)$ | $0$ |
$72$ | $6$ | $(1,9,8,7,4,3)(2,5,6)$ | $1$ |
$108$ | $6$ | $(1,4)(2,7,5,9,6,3)$ | $0$ |
$216$ | $6$ | $(1,2,3,4,5,7)(6,9,8)$ | $0$ |
$144$ | $9$ | $(1,2,7,4,5,9,8,6,3)$ | $-1$ |
$108$ | $12$ | $(1,7,4,3)(2,5,6)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.