Basic invariants
Dimension: | $12$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(468\!\cdots\!048\)\(\medspace = 2^{10} \cdot 3^{8} \cdot 587^{5} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.155337218304.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 36T2214 |
Parity: | odd |
Determinant: | 1.587.2t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.1.155337218304.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 3x^{8} - 4x^{6} + 18x^{5} + 6x^{4} - 20x^{3} - 8x^{2} + x - 3 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{3} + x + 40 \)
Roots:
$r_{ 1 }$ | $=$ | \( 26 a^{2} + 25 a + 1 + \left(24 a^{2} + 24 a + 42\right)\cdot 43 + \left(41 a^{2} + 4 a + 24\right)\cdot 43^{2} + \left(23 a^{2} + 41 a + 38\right)\cdot 43^{3} + \left(21 a^{2} + 5 a + 1\right)\cdot 43^{4} + \left(37 a^{2} + 38 a + 19\right)\cdot 43^{5} + \left(27 a^{2} + 21 a + 35\right)\cdot 43^{6} + \left(35 a^{2} + 4 a + 36\right)\cdot 43^{7} + \left(24 a^{2} + 26 a + 42\right)\cdot 43^{8} + \left(4 a^{2} + 18 a + 33\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 2 }$ | $=$ | \( 37 a^{2} + 5 a + 37 + \left(32 a^{2} + 27 a + 18\right)\cdot 43 + \left(17 a^{2} + 12 a + 23\right)\cdot 43^{2} + \left(22 a^{2} + 16 a + 37\right)\cdot 43^{3} + \left(27 a^{2} + 18 a + 5\right)\cdot 43^{4} + \left(12 a^{2} + 10 a + 31\right)\cdot 43^{5} + \left(3 a^{2} + 7 a + 4\right)\cdot 43^{6} + \left(13 a^{2} + 10 a + 36\right)\cdot 43^{7} + \left(6 a^{2} + 29 a + 1\right)\cdot 43^{8} + \left(29 a^{2} + 3 a + 36\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 3 }$ | $=$ | \( a^{2} + 20 a + 9 + \left(15 a^{2} + 35 a + 32\right)\cdot 43 + \left(25 a^{2} + 25 a + 23\right)\cdot 43^{2} + \left(34 a^{2} + 36 a + 6\right)\cdot 43^{3} + \left(26 a^{2} + 25 a + 2\right)\cdot 43^{4} + \left(37 a^{2} + 7 a + 15\right)\cdot 43^{5} + \left(9 a^{2} + 2 a + 35\right)\cdot 43^{6} + \left(32 a^{2} + 17 a + 26\right)\cdot 43^{7} + \left(32 a^{2} + 29 a + 42\right)\cdot 43^{8} + \left(36 a^{2} + 39 a + 36\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 4 }$ | $=$ | \( 32 a^{2} + 2 a + 16 + \left(10 a^{2} + 22 a + 2\right)\cdot 43 + \left(11 a^{2} + 25 a + 32\right)\cdot 43^{2} + \left(37 a^{2} + 16 a + 18\right)\cdot 43^{3} + \left(12 a^{2} + 23 a + 22\right)\cdot 43^{4} + \left(34 a^{2} + 4 a + 24\right)\cdot 43^{5} + \left(31 a^{2} + 14 a + 18\right)\cdot 43^{6} + \left(38 a^{2} + 6 a + 7\right)\cdot 43^{7} + \left(6 a^{2} + 27 a + 29\right)\cdot 43^{8} + \left(37 a^{2} + 31 a + 38\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 5 }$ | $=$ | \( 23 a^{2} + 13 a + 42 + \left(28 a^{2} + 34 a + 15\right)\cdot 43 + \left(26 a^{2} + 25 a + 29\right)\cdot 43^{2} + \left(39 a^{2} + 28 a + 34\right)\cdot 43^{3} + \left(36 a^{2} + 18 a + 40\right)\cdot 43^{4} + \left(35 a^{2} + 37 a + 17\right)\cdot 43^{5} + \left(11 a^{2} + 13 a + 10\right)\cdot 43^{6} + \left(37 a^{2} + 28 a + 9\right)\cdot 43^{7} + \left(11 a^{2} + 30 a + 34\right)\cdot 43^{8} + \left(9 a^{2} + 20 a + 22\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 6 }$ | $=$ | \( 28 a^{2} + 16 a + 27 + \left(7 a^{2} + 39 a + 41\right)\cdot 43 + \left(33 a^{2} + 12 a + 28\right)\cdot 43^{2} + \left(24 a^{2} + 28 a + 28\right)\cdot 43^{3} + \left(8 a^{2} + 13 a + 18\right)\cdot 43^{4} + \left(14 a^{2} + 42\right)\cdot 43^{5} + \left(26 a^{2} + 7 a + 31\right)\cdot 43^{6} + \left(11 a^{2} + 32 a + 41\right)\cdot 43^{7} + \left(11 a^{2} + 32 a + 13\right)\cdot 43^{8} + \left(a^{2} + 35 a + 13\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{2} + 7 a + 32 + \left(20 a^{2} + 11 a + 35\right)\cdot 43 + \left(27 a^{2} + 4 a + 10\right)\cdot 43^{2} + \left(26 a^{2} + 21 a + 1\right)\cdot 43^{3} + \left(7 a^{2} + 3 a + 18\right)\cdot 43^{4} + \left(34 a^{2} + 35 a + 41\right)\cdot 43^{5} + \left(6 a^{2} + 33 a + 18\right)\cdot 43^{6} + \left(42 a^{2} + 36 a + 33\right)\cdot 43^{7} + \left(41 a^{2} + 23 a + 5\right)\cdot 43^{8} + \left(4 a^{2} + 10 a + 30\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 8 }$ | $=$ | \( 19 a^{2} + 10 a + 36 + \left(42 a^{2} + 16 a + 37\right)\cdot 43 + \left(33 a^{2} + 34 a + 32\right)\cdot 43^{2} + \left(11 a^{2} + 20 a + 1\right)\cdot 43^{3} + \left(22 a^{2} + 41 a\right)\cdot 43^{4} + \left(12 a^{2} + 40 a + 10\right)\cdot 43^{5} + \left(21 a^{2} + 26 a + 40\right)\cdot 43^{6} + \left(16 a^{2} + 40 a + 6\right)\cdot 43^{7} + \left(41 a^{2} + 25 a + 9\right)\cdot 43^{8} + \left(39 a^{2} + 25 a + 26\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 9 }$ | $=$ | \( 35 a^{2} + 31 a + 18 + \left(32 a^{2} + 4 a + 31\right)\cdot 43 + \left(40 a^{2} + 26 a + 8\right)\cdot 43^{2} + \left(36 a^{2} + 5 a + 4\right)\cdot 43^{3} + \left(7 a^{2} + 21 a + 19\right)\cdot 43^{4} + \left(39 a^{2} + 40 a + 13\right)\cdot 43^{5} + \left(32 a^{2} + a + 19\right)\cdot 43^{6} + \left(30 a^{2} + 39 a + 16\right)\cdot 43^{7} + \left(37 a^{2} + 32 a + 35\right)\cdot 43^{8} + \left(8 a^{2} + 28 a + 19\right)\cdot 43^{9} +O(43^{10})\) |
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 |
$1$ | $1$ | $()$ | $12$ |
$9$ | $2$ | $(2,7)$ | $-4$ |
$18$ | $2$ | $(1,2)(4,7)(6,9)$ | $2$ |
$27$ | $2$ | $(1,4)(2,7)(3,5)$ | $0$ |
$27$ | $2$ | $(1,4)(2,7)$ | $0$ |
$54$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
$6$ | $3$ | $(3,5,8)$ | $0$ |
$8$ | $3$ | $(1,4,6)(2,7,9)(3,5,8)$ | $3$ |
$12$ | $3$ | $(1,4,6)(3,5,8)$ | $-3$ |
$72$ | $3$ | $(1,3,2)(4,5,7)(6,8,9)$ | $0$ |
$54$ | $4$ | $(1,2,4,7)(6,9)$ | $0$ |
$162$ | $4$ | $(2,5,7,3)(4,6)(8,9)$ | $0$ |
$36$ | $6$ | $(1,2)(3,5,8)(4,7)(6,9)$ | $2$ |
$36$ | $6$ | $(2,3,7,5,9,8)$ | $-1$ |
$36$ | $6$ | $(2,7)(3,5,8)$ | $2$ |
$36$ | $6$ | $(1,4,6)(2,7)(3,5,8)$ | $-1$ |
$54$ | $6$ | $(1,4)(2,7)(3,8,5)$ | $0$ |
$72$ | $6$ | $(1,7,4,9,6,2)(3,5,8)$ | $-1$ |
$108$ | $6$ | $(1,3,4,5,6,8)(2,7)$ | $1$ |
$216$ | $6$ | $(1,3,2,4,5,7)(6,8,9)$ | $0$ |
$144$ | $9$ | $(1,3,7,4,5,9,6,8,2)$ | $0$ |
$108$ | $12$ | $(1,2,4,7)(3,5,8)(6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.