Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(73923622208674101504\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 271^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.3.412698468096.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 24T2893 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.3.412698468096.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 3x^{8} + 6x^{7} - 6x^{6} + 4x^{5} - 10x^{3} + 10x^{2} - 9x - 3 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{3} + 6x + 35 \)
Roots:
$r_{ 1 }$ | $=$ | \( 17 a^{2} + a + 2 + \left(13 a^{2} + 31 a + 35\right)\cdot 37 + \left(24 a^{2} + 30 a + 7\right)\cdot 37^{2} + \left(17 a^{2} + 4 a + 3\right)\cdot 37^{3} + \left(32 a^{2} + 22 a + 7\right)\cdot 37^{4} + \left(4 a^{2} + 9 a + 35\right)\cdot 37^{5} + \left(13 a^{2} + 23 a + 10\right)\cdot 37^{6} + \left(17 a^{2} + 18 a + 15\right)\cdot 37^{7} + \left(18 a^{2} + 30 a + 32\right)\cdot 37^{8} + \left(11 a^{2} + 20 a + 25\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 2 }$ | $=$ | \( 6 a + 21 + \left(23 a^{2} + a + 1\right)\cdot 37 + \left(12 a^{2} + 6 a + 27\right)\cdot 37^{2} + \left(11 a^{2} + 18 a + 24\right)\cdot 37^{3} + \left(a^{2} + 24 a + 1\right)\cdot 37^{4} + \left(13 a^{2} + 23 a + 30\right)\cdot 37^{5} + \left(25 a^{2} + 33 a + 3\right)\cdot 37^{6} + \left(28 a^{2} + 21\right)\cdot 37^{7} + \left(31 a^{2} + 5 a + 17\right)\cdot 37^{8} + \left(16 a^{2} + 12 a + 15\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 3 }$ | $=$ | \( 30 a^{2} + 14 a + 18 + \left(7 a^{2} + 4 a + 29\right)\cdot 37 + \left(25 a^{2} + 22 a + 28\right)\cdot 37^{2} + \left(15 a^{2} + 4 a + 2\right)\cdot 37^{3} + \left(27 a^{2} + 14 a + 14\right)\cdot 37^{4} + \left(7 a^{2} + 7 a\right)\cdot 37^{5} + \left(33 a^{2} + 4 a + 13\right)\cdot 37^{6} + \left(20 a^{2} + 35 a + 9\right)\cdot 37^{7} + \left(8 a^{2} + 26 a\right)\cdot 37^{8} + \left(15 a^{2} + 32 a + 22\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 4 }$ | $=$ | \( 5 a^{2} + 18 a + 4 + \left(27 a^{2} + 18\right)\cdot 37 + \left(36 a^{2} + 24 a + 12\right)\cdot 37^{2} + \left(25 a^{2} + 26 a + 9\right)\cdot 37^{3} + \left(4 a^{2} + 24 a + 15\right)\cdot 37^{4} + \left(2 a^{2} + 35 a + 23\right)\cdot 37^{5} + \left(21 a^{2} + 34 a + 23\right)\cdot 37^{6} + \left(2 a^{2} + 5 a + 27\right)\cdot 37^{7} + \left(19 a^{2} + a + 3\right)\cdot 37^{8} + \left(13 a^{2} + 24 a + 2\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 5 }$ | $=$ | \( 29 a^{2} + 5 a + 14 + \left(32 a^{2} + 27 a + 18\right)\cdot 37 + \left(35 a^{2} + 32 a + 34\right)\cdot 37^{2} + \left(27 a^{2} + 26 a + 14\right)\cdot 37^{3} + \left(9 a^{2} + 32 a + 17\right)\cdot 37^{4} + \left(36 a^{2} + 3\right)\cdot 37^{5} + \left(17 a + 32\right)\cdot 37^{6} + \left(36 a^{2} + 26 a + 32\right)\cdot 37^{7} + \left(28 a^{2} + 4 a + 7\right)\cdot 37^{8} + \left(9 a^{2} + 12 a\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 6 }$ | $=$ | \( 7 a^{2} + 17 a + 36 + \left(6 a^{2} + 31 a + 5\right)\cdot 37 + \left(36 a^{2} + 8 a + 18\right)\cdot 37^{2} + \left(9 a^{2} + 14 a + 9\right)\cdot 37^{3} + \left(8 a^{2} + 35 a + 21\right)\cdot 37^{4} + \left(16 a^{2} + 5 a + 6\right)\cdot 37^{5} + \left(15 a^{2} + 36 a + 20\right)\cdot 37^{6} + \left(24 a^{2} + 6\right)\cdot 37^{7} + \left(33 a^{2} + 5 a + 19\right)\cdot 37^{8} + \left(4 a^{2} + 29 a + 36\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 7 }$ | $=$ | \( 15 a^{2} + 18 a + 32 + \left(33 a^{2} + 5 a + 20\right)\cdot 37 + \left(12 a^{2} + 19 a + 16\right)\cdot 37^{2} + \left(30 a^{2} + 5 a + 24\right)\cdot 37^{3} + \left(36 a^{2} + 27 a + 14\right)\cdot 37^{4} + \left(29 a^{2} + 28 a + 15\right)\cdot 37^{5} + \left(2 a^{2} + 15 a + 2\right)\cdot 37^{6} + \left(17 a^{2} + 12 a + 31\right)\cdot 37^{7} + \left(36 a^{2} + 5 a\right)\cdot 37^{8} + \left(11 a^{2} + 29 a + 9\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 8 }$ | $=$ | \( 13 a^{2} + 19 a + 23 + \left(17 a^{2} + 11 a + 13\right)\cdot 37 + \left(13 a^{2} + 34 a + 1\right)\cdot 37^{2} + \left(9 a^{2} + 17 a + 7\right)\cdot 37^{3} + \left(33 a^{2} + 16 a + 10\right)\cdot 37^{4} + \left(15 a^{2} + 21 a + 5\right)\cdot 37^{5} + \left(8 a^{2} + 14 a + 29\right)\cdot 37^{6} + \left(32 a^{2} + 17 a\right)\cdot 37^{7} + \left(21 a^{2} + a + 9\right)\cdot 37^{8} + \left(20 a^{2} + 24 a + 25\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 9 }$ | $=$ | \( 32 a^{2} + 13 a + 1 + \left(23 a^{2} + 35 a + 5\right)\cdot 37 + \left(24 a^{2} + 6 a + 1\right)\cdot 37^{2} + \left(36 a^{2} + 29 a + 15\right)\cdot 37^{3} + \left(30 a^{2} + 24 a + 9\right)\cdot 37^{4} + \left(21 a^{2} + 14 a + 28\right)\cdot 37^{5} + \left(27 a^{2} + 5 a + 12\right)\cdot 37^{6} + \left(5 a^{2} + 30 a + 3\right)\cdot 37^{7} + \left(23 a^{2} + 30 a + 20\right)\cdot 37^{8} + \left(6 a^{2} + 11\right)\cdot 37^{9} +O(37^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(1,4)$ | $0$ |
$18$ | $2$ | $(1,2)(3,4)(6,7)$ | $-4$ |
$27$ | $2$ | $(1,4)(2,3)(5,8)$ | $0$ |
$27$ | $2$ | $(1,4)(2,3)$ | $0$ |
$54$ | $2$ | $(1,4)(2,5)(3,8)(6,9)$ | $0$ |
$6$ | $3$ | $(5,8,9)$ | $-4$ |
$8$ | $3$ | $(1,4,7)(2,3,6)(5,8,9)$ | $-1$ |
$12$ | $3$ | $(1,4,7)(5,8,9)$ | $2$ |
$72$ | $3$ | $(1,2,5)(3,8,4)(6,9,7)$ | $2$ |
$54$ | $4$ | $(1,3,4,2)(6,7)$ | $0$ |
$162$ | $4$ | $(1,8,4,5)(3,6)(7,9)$ | $0$ |
$36$ | $6$ | $(1,2)(3,4)(5,8,9)(6,7)$ | $2$ |
$36$ | $6$ | $(1,5,4,8,7,9)$ | $2$ |
$36$ | $6$ | $(1,4)(5,8,9)$ | $0$ |
$36$ | $6$ | $(1,4)(2,3,6)(5,8,9)$ | $0$ |
$54$ | $6$ | $(1,4)(2,3)(5,9,8)$ | $0$ |
$72$ | $6$ | $(1,2,4,3,7,6)(5,8,9)$ | $-1$ |
$108$ | $6$ | $(1,4)(2,5,3,8,6,9)$ | $0$ |
$216$ | $6$ | $(1,3,8,4,2,5)(6,9,7)$ | $0$ |
$144$ | $9$ | $(1,2,5,4,3,8,7,6,9)$ | $-1$ |
$108$ | $12$ | $(1,3,4,2)(5,8,9)(6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.