Basic invariants
Dimension: | $6$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(336251692\)\(\medspace = 2^{2} \cdot 7 \cdot 229^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.5.77001637468.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T319 |
Parity: | even |
Determinant: | 1.6412.2t1.a.a |
Projective image: | $S_3\wr S_3$ |
Projective stem field: | Galois closure of 9.5.77001637468.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 4x^{8} + 2x^{7} + 7x^{6} - 9x^{5} + 6x^{4} + x^{3} - 5x^{2} + 11x + 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a^{3} + 13 a^{2} + 22 a + 9 + \left(15 a^{3} + 8 a^{2} + 13\right)\cdot 23 + \left(9 a^{3} + 5 a^{2} + 11 a + 3\right)\cdot 23^{2} + \left(17 a^{3} + 8 a^{2} + 11 a + 19\right)\cdot 23^{3} + \left(16 a^{3} + 22 a^{2} + 20 a + 3\right)\cdot 23^{4} + \left(4 a^{3} + 20 a^{2} + 21 a + 7\right)\cdot 23^{5} + \left(9 a^{3} + 11 a^{2} + 22 a + 10\right)\cdot 23^{6} + \left(a^{3} + 12 a^{2} + 15 a + 2\right)\cdot 23^{7} + \left(6 a^{3} + 2 a^{2} + 6 a + 19\right)\cdot 23^{8} + \left(a^{3} + 5 a^{2} + 3 a\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 2 }$ | $=$ | \( 9 a^{3} + 21 a^{2} + 12 a + 18 + \left(4 a^{3} + 19 a^{2} + 7 a\right)\cdot 23 + \left(21 a^{3} + 8 a^{2} + 12\right)\cdot 23^{2} + \left(11 a^{3} + 10 a^{2} + 16 a + 1\right)\cdot 23^{3} + \left(15 a^{3} + 4 a^{2} + 15 a + 11\right)\cdot 23^{4} + \left(6 a^{3} + 20 a^{2} + 17 a + 16\right)\cdot 23^{5} + \left(17 a^{3} + 10 a^{2} + 19 a + 14\right)\cdot 23^{6} + \left(a^{3} + 12 a^{2} + 7 a + 1\right)\cdot 23^{7} + \left(a^{3} + 4 a^{2} + 22 a + 20\right)\cdot 23^{8} + \left(7 a^{2} + 10 a + 3\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 3 }$ | $=$ | \( 9 a^{3} + 18 a^{2} + 18 a + 22 + \left(16 a^{3} + 14 a^{2} + 21 a + 8\right)\cdot 23 + \left(10 a^{3} + 4 a + 8\right)\cdot 23^{2} + \left(11 a^{3} + 15 a^{2} + 19 a + 6\right)\cdot 23^{3} + \left(22 a^{2} + 17 a + 17\right)\cdot 23^{4} + \left(17 a^{3} + 3 a^{2} + 13 a + 6\right)\cdot 23^{5} + \left(12 a^{3} + 7 a + 3\right)\cdot 23^{6} + \left(13 a^{3} + 3 a^{2} + 7 a + 13\right)\cdot 23^{7} + \left(15 a^{3} + 14 a + 20\right)\cdot 23^{8} + \left(20 a^{3} + 6 a^{2} + 3 a + 22\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 4 }$ | $=$ | \( 22 a^{3} + 21 a^{2} + 21 a + 16 + \left(10 a^{3} + 12 a + 21\right)\cdot 23 + \left(16 a^{3} + 2 a^{2} + a + 10\right)\cdot 23^{2} + \left(9 a^{3} + 13 a^{2} + 5 a + 16\right)\cdot 23^{3} + \left(20 a^{3} + 19 a^{2} + 4 a + 13\right)\cdot 23^{4} + \left(19 a^{3} + 5 a^{2} + 4 a + 8\right)\cdot 23^{5} + \left(22 a^{3} + 5 a^{2} + 5 a + 2\right)\cdot 23^{6} + \left(a^{3} + 22 a^{2} + 15\right)\cdot 23^{7} + \left(21 a^{3} + 18 a^{2} + 13 a + 13\right)\cdot 23^{8} + \left(4 a^{3} + 3 a^{2} + 10 a + 4\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 5 }$ | $=$ | \( a^{3} + 2 a^{2} + 2 a + 16 + \left(12 a^{3} + 22 a^{2} + 10 a + 5\right)\cdot 23 + \left(6 a^{3} + 20 a^{2} + 21 a + 18\right)\cdot 23^{2} + \left(13 a^{3} + 9 a^{2} + 17 a + 10\right)\cdot 23^{3} + \left(2 a^{3} + 3 a^{2} + 18 a + 5\right)\cdot 23^{4} + \left(3 a^{3} + 17 a^{2} + 18 a + 22\right)\cdot 23^{5} + \left(17 a^{2} + 17 a + 1\right)\cdot 23^{6} + \left(21 a^{3} + 22 a + 18\right)\cdot 23^{7} + \left(a^{3} + 4 a^{2} + 9 a + 22\right)\cdot 23^{8} + \left(18 a^{3} + 19 a^{2} + 12 a + 13\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 6 }$ | $=$ | \( 4 a^{3} + 19 a^{2} + 16 a + 7 + \left(17 a^{3} + a^{2} + 3 a + 6\right)\cdot 23 + \left(16 a^{3} + 20 a^{2} + 5 a + 17\right)\cdot 23^{2} + \left(a^{3} + 7 a^{2} + 12 a + 7\right)\cdot 23^{3} + \left(2 a^{3} + 9 a^{2} + 20 a + 22\right)\cdot 23^{4} + \left(21 a^{3} + 13 a^{2} + 18 a + 21\right)\cdot 23^{5} + \left(16 a^{2} + 19\right)\cdot 23^{6} + \left(7 a^{3} + a^{2} + 17 a + 14\right)\cdot 23^{7} + \left(6 a^{3} + 4 a^{2} + 8 a + 7\right)\cdot 23^{8} + \left(17 a^{3} + 16 a^{2} + 22 a + 21\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 7 }$ | $=$ | \( 14 a^{3} + 5 a^{2} + 5 a + 22 + \left(6 a^{3} + 8 a^{2} + a + 14\right)\cdot 23 + \left(12 a^{3} + 22 a^{2} + 18 a + 11\right)\cdot 23^{2} + \left(11 a^{3} + 7 a^{2} + 3 a + 2\right)\cdot 23^{3} + \left(22 a^{3} + 5 a + 4\right)\cdot 23^{4} + \left(5 a^{3} + 19 a^{2} + 9 a + 4\right)\cdot 23^{5} + \left(10 a^{3} + 22 a^{2} + 15 a + 19\right)\cdot 23^{6} + \left(9 a^{3} + 19 a^{2} + 15 a + 8\right)\cdot 23^{7} + \left(7 a^{3} + 22 a^{2} + 8 a + 1\right)\cdot 23^{8} + \left(2 a^{3} + 16 a^{2} + 19 a + 2\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 8 }$ | $=$ | \( 2 a^{3} + 16 a^{2} + 19 a + 20 + \left(9 a^{3} + 15 a^{2} + 10 a + 3\right)\cdot 23 + \left(21 a^{3} + 11 a^{2} + 6 a + 19\right)\cdot 23^{2} + \left(14 a^{3} + 19 a^{2} + 6 a + 17\right)\cdot 23^{3} + \left(11 a^{3} + 9 a^{2} + 12 a + 15\right)\cdot 23^{4} + \left(13 a^{3} + 14 a^{2} + 10 a + 1\right)\cdot 23^{5} + \left(18 a^{3} + 6 a^{2} + 2 a + 21\right)\cdot 23^{6} + \left(12 a^{3} + 19 a^{2} + 5 a + 19\right)\cdot 23^{7} + \left(9 a^{3} + 11 a^{2} + 8 a + 13\right)\cdot 23^{8} + \left(4 a^{3} + 17 a^{2} + 9 a + 12\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 9 }$ | $=$ | \( 12 + 16\cdot 23 + 13\cdot 23^{2} + 9\cdot 23^{3} + 21\cdot 23^{4} + 2\cdot 23^{5} + 22\cdot 23^{6} + 20\cdot 23^{7} + 18\cdot 23^{8} + 9\cdot 23^{9} +O(23^{10})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$9$ | $2$ | $(3,7)$ | $4$ |
$18$ | $2$ | $(3,4)(6,7)(8,9)$ | $-2$ |
$27$ | $2$ | $(3,7)(4,6)$ | $2$ |
$27$ | $2$ | $(1,2)(3,7)(4,6)$ | $0$ |
$54$ | $2$ | $(1,3)(2,7)(4,6)(5,9)$ | $0$ |
$6$ | $3$ | $(1,2,5)$ | $3$ |
$8$ | $3$ | $(1,2,5)(3,7,9)(4,6,8)$ | $-3$ |
$12$ | $3$ | $(1,2,5)(3,7,9)$ | $0$ |
$72$ | $3$ | $(1,3,4)(2,7,6)(5,9,8)$ | $0$ |
$54$ | $4$ | $(3,4,7,6)(8,9)$ | $-2$ |
$162$ | $4$ | $(1,3,2,7)(4,6)(5,9)$ | $0$ |
$36$ | $6$ | $(1,2,5)(3,4)(6,7)(8,9)$ | $1$ |
$36$ | $6$ | $(1,7,2,9,5,3)$ | $-2$ |
$36$ | $6$ | $(1,2,5)(3,7)$ | $1$ |
$36$ | $6$ | $(1,2,5)(3,7)(4,6,8)$ | $-2$ |
$54$ | $6$ | $(1,5,2)(3,7)(4,6)$ | $-1$ |
$72$ | $6$ | $(1,2,5)(3,4,9,8,7,6)$ | $1$ |
$108$ | $6$ | $(1,7,2,9,5,3)(4,6)$ | $0$ |
$216$ | $6$ | $(1,3,6,2,7,4)(5,9,8)$ | $0$ |
$144$ | $9$ | $(1,7,6,2,9,8,5,3,4)$ | $0$ |
$108$ | $12$ | $(1,2,5)(3,4,7,6)(8,9)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.