Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(164648481361000000\)\(\medspace = 2^{6} \cdot 5^{6} \cdot 7^{8} \cdot 13^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.1529437000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr C_3$ |
| Projective stem field: | Galois closure of 9.5.1529437000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} + x^{7} - 3x^{6} - 23x^{5} + 68x^{4} - 124x^{3} + 93x^{2} + x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{3} + 2x + 18 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 22 a^{2} + 3 a + 2 + \left(21 a^{2} + 4 a + 11\right)\cdot 23 + \left(12 a^{2} + 19 a + 4\right)\cdot 23^{2} + \left(14 a^{2} + 16 a + 6\right)\cdot 23^{3} + \left(2 a^{2} + 8 a + 5\right)\cdot 23^{4} + \left(22 a^{2} + 19 a + 21\right)\cdot 23^{5} + \left(9 a^{2} + 21 a + 20\right)\cdot 23^{6} + \left(6 a^{2} + 9\right)\cdot 23^{7} + \left(4 a^{2} + 18 a + 7\right)\cdot 23^{8} + \left(10 a^{2} + 8 a + 21\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 10 a + 17 + \left(20 a^{2} + 21 a + 17\right)\cdot 23 + \left(20 a^{2} + 21 a + 10\right)\cdot 23^{2} + \left(11 a^{2} + 22 a + 1\right)\cdot 23^{3} + \left(6 a^{2} + a + 20\right)\cdot 23^{4} + \left(11 a^{2} + 14 a + 22\right)\cdot 23^{5} + \left(3 a^{2} + 2 a + 16\right)\cdot 23^{6} + \left(4 a^{2} + 21 a + 10\right)\cdot 23^{7} + \left(22 a^{2} + 9 a + 19\right)\cdot 23^{8} + \left(6 a^{2} + 21 a\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 18 a^{2} + 5 a + 18 + 22\cdot 23 + \left(5 a^{2} + 3 a + 4\right)\cdot 23^{2} + \left(3 a^{2} + 19 a + 5\right)\cdot 23^{3} + \left(7 a^{2} + 2 a + 13\right)\cdot 23^{4} + \left(16 a^{2} + 4 a + 6\right)\cdot 23^{5} + \left(21 a^{2} + 18\right)\cdot 23^{6} + \left(4 a^{2} + 14 a + 11\right)\cdot 23^{7} + \left(22 a^{2} + 2 a + 19\right)\cdot 23^{8} + \left(10 a^{2} + 18 a + 13\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 4 a^{2} + 2 a + 8 + \left(13 a^{2} + 16 a + 6\right)\cdot 23 + \left(4 a^{2} + 3 a + 5\right)\cdot 23^{2} + \left(17 a^{2} + 19 a + 12\right)\cdot 23^{3} + \left(20 a^{2} + 21 a + 14\right)\cdot 23^{4} + \left(9 a^{2} + a + 13\right)\cdot 23^{5} + \left(9 a^{2} + 10 a + 15\right)\cdot 23^{6} + \left(a^{2} + 16 a + 10\right)\cdot 23^{7} + \left(17 a^{2} + 9 a\right)\cdot 23^{8} + \left(8 a^{2} + 20 a + 20\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a^{2} + 8 a + 16 + \left(2 a^{2} + a + 1\right)\cdot 23 + \left(20 a^{2} + 21 a + 2\right)\cdot 23^{2} + \left(7 a^{2} + 3 a + 19\right)\cdot 23^{3} + \left(9 a^{2} + 18 a\right)\cdot 23^{4} + \left(18 a^{2} + 4 a + 17\right)\cdot 23^{5} + \left(20 a^{2} + 20 a + 1\right)\cdot 23^{6} + \left(13 a^{2} + 10 a + 16\right)\cdot 23^{7} + \left(a^{2} + 10 a + 22\right)\cdot 23^{8} + \left(5 a^{2} + 6 a + 5\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 3 a^{2} + 6 a + 22 + \left(20 a^{2} + 18 a + 7\right)\cdot 23 + \left(11 a^{2} + 13 a + 7\right)\cdot 23^{2} + \left(a^{2} + 2 a + 14\right)\cdot 23^{3} + \left(13 a^{2} + 17 a + 19\right)\cdot 23^{4} + \left(3 a^{2} + 6 a + 12\right)\cdot 23^{5} + \left(4 a^{2} + 21 a + 8\right)\cdot 23^{6} + \left(4 a^{2} + 19 a + 14\right)\cdot 23^{7} + \left(8 a^{2} + 15 a + 11\right)\cdot 23^{8} + \left(19 a^{2} + 20 a + 3\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 21 a + 11 + \left(20 a^{2} + 11 a + 8\right)\cdot 23 + \left(17 a^{2} + 3 a + 3\right)\cdot 23^{2} + \left(19 a^{2} + 5 a + 13\right)\cdot 23^{3} + \left(18 a^{2} + 5 a + 11\right)\cdot 23^{4} + \left(13 a^{2} + 14 a + 2\right)\cdot 23^{5} + \left(14 a^{2} + 3 a + 4\right)\cdot 23^{6} + \left(15 a^{2} + 6 a + 22\right)\cdot 23^{7} + \left(7 a^{2} + 11 a + 11\right)\cdot 23^{8} + \left(8 a^{2} + 12 a + 3\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 8 }$ |
$=$ |
\( a^{2} + 22 a + 20 + \left(4 a^{2} + 6 a + 17\right)\cdot 23 + \left(15 a^{2} + 22\right)\cdot 23^{2} + \left(11 a^{2} + a + 9\right)\cdot 23^{3} + \left(a^{2} + 9 a + 11\right)\cdot 23^{4} + \left(10 a^{2} + 12 a + 20\right)\cdot 23^{5} + \left(21 a^{2} + 20 a + 20\right)\cdot 23^{6} + \left(15 a + 17\right)\cdot 23^{7} + \left(11 a^{2} + 16 a + 8\right)\cdot 23^{8} + \left(4 a^{2} + a + 21\right)\cdot 23^{9} +O(23^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 16 a^{2} + 15 a + 1 + \left(12 a^{2} + 11 a + 21\right)\cdot 23 + \left(6 a^{2} + 5 a + 7\right)\cdot 23^{2} + \left(4 a^{2} + a + 10\right)\cdot 23^{3} + \left(12 a^{2} + 7 a + 18\right)\cdot 23^{4} + \left(9 a^{2} + 14 a + 20\right)\cdot 23^{5} + \left(9 a^{2} + 14 a + 7\right)\cdot 23^{6} + \left(17 a^{2} + 9 a + 1\right)\cdot 23^{7} + \left(20 a^{2} + 20 a + 13\right)\cdot 23^{8} + \left(17 a^{2} + 4 a + 1\right)\cdot 23^{9} +O(23^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(1,2)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,7)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(7,9)$ | $0$ |
| $6$ | $3$ | $(5,6,8)$ | $0$ |
| $8$ | $3$ | $(1,2,4)(3,7,9)(5,6,8)$ | $3$ |
| $12$ | $3$ | $(3,7,9)(5,6,8)$ | $-3$ |
| $36$ | $3$ | $(1,5,3)(2,6,7)(4,8,9)$ | $0$ |
| $36$ | $3$ | $(1,3,5)(2,7,6)(4,9,8)$ | $0$ |
| $18$ | $6$ | $(1,2)(5,6,8)$ | $-2$ |
| $18$ | $6$ | $(1,2)(3,7,9)$ | $-2$ |
| $36$ | $6$ | $(1,2)(3,7,9)(5,6,8)$ | $1$ |
| $54$ | $6$ | $(1,2)(5,6,8)(7,9)$ | $0$ |
| $108$ | $6$ | $(1,6,7,2,5,3)(4,8,9)$ | $0$ |
| $108$ | $6$ | $(1,3,5,2,7,6)(4,9,8)$ | $0$ |
| $72$ | $9$ | $(1,5,7,2,6,9,4,8,3)$ | $0$ |
| $72$ | $9$ | $(1,7,6,4,3,5,2,9,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.