Basic invariants
Dimension: | $12$ |
Group: | $S_3 \wr C_3 $ |
Conductor: | \(331258496613663609\)\(\medspace = 3^{26} \cdot 19^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.5.22082967873.1 |
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 9.5.22082967873.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 9x^{7} + 27x^{5} - 36x^{3} + 27x - 9 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 10 a^{2} + 8 a + 3 + \left(10 a^{2} + 5 a + 25\right)\cdot 29 + \left(15 a^{2} + 21\right)\cdot 29^{2} + \left(27 a^{2} + 22 a + 4\right)\cdot 29^{3} + \left(19 a^{2} + 18 a + 1\right)\cdot 29^{4} + \left(20 a^{2} + 27 a + 14\right)\cdot 29^{5} + \left(27 a^{2} + 28 a + 28\right)\cdot 29^{6} + \left(10 a^{2} + 11 a + 26\right)\cdot 29^{7} + \left(18 a^{2} + 6 a + 8\right)\cdot 29^{8} + \left(15 a^{2} + 20 a + 18\right)\cdot 29^{9} +O(29^{10})\)
$r_{ 2 }$ |
$=$ |
\( 23 a^{2} + 24 a + 3 + \left(23 a^{2} + 26 a + 9\right)\cdot 29 + \left(28 a^{2} + 4 a + 18\right)\cdot 29^{2} + \left(22 a^{2} + 28\right)\cdot 29^{3} + \left(2 a^{2} + 23 a + 8\right)\cdot 29^{4} + \left(13 a^{2} + a + 28\right)\cdot 29^{5} + \left(27 a^{2} + 7 a + 5\right)\cdot 29^{6} + \left(25 a^{2} + 11 a + 23\right)\cdot 29^{7} + \left(19 a^{2} + 8 a + 26\right)\cdot 29^{8} + \left(14 a^{2} + 22 a + 5\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 7 a^{2} + 22 a + 28 + \left(4 a^{2} + 16\right)\cdot 29 + \left(7 a^{2} + 25 a + 20\right)\cdot 29^{2} + \left(14 a^{2} + 24 a + 25\right)\cdot 29^{3} + \left(3 a^{2} + 16 a + 17\right)\cdot 29^{4} + \left(27 a^{2} + 27 a + 22\right)\cdot 29^{5} + \left(18 a^{2} + 23 a + 16\right)\cdot 29^{6} + \left(8 a^{2} + 8 a + 4\right)\cdot 29^{7} + \left(10 a^{2} + 7 a + 27\right)\cdot 29^{8} + \left(10 a^{2} + 22 a + 20\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 12 a^{2} + 28 a + 25 + \left(14 a^{2} + 22 a + 20\right)\cdot 29 + \left(6 a^{2} + 3 a + 19\right)\cdot 29^{2} + \left(16 a^{2} + 11 a + 18\right)\cdot 29^{3} + \left(5 a^{2} + 22 a + 20\right)\cdot 29^{4} + \left(10 a^{2} + 2 a + 9\right)\cdot 29^{5} + \left(11 a^{2} + 5 a + 16\right)\cdot 29^{6} + \left(9 a^{2} + 8 a + 5\right)\cdot 29^{7} + \left(15 a + 4\right)\cdot 29^{8} + \left(3 a^{2} + 15 a + 11\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 21 a^{2} + 16 a + 10 + \left(7 a^{2} + 27 a + 26\right)\cdot 29 + \left(21 a^{2} + 11 a + 17\right)\cdot 29^{2} + \left(21 a^{2} + 15 a + 7\right)\cdot 29^{3} + \left(28 a^{2} + 5 a + 24\right)\cdot 29^{4} + \left(24 a^{2} + 26 a + 24\right)\cdot 29^{5} + \left(a + 18\right)\cdot 29^{6} + \left(9 a + 17\right)\cdot 29^{7} + \left(13 a^{2} + 8 a + 17\right)\cdot 29^{8} + \left(14 a^{2} + 21 a + 5\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 5 a^{2} + 3 a + 6 + \left(21 a^{2} + 20 a + 20\right)\cdot 29 + \left(5 a^{2} + 16 a + 16\right)\cdot 29^{2} + \left(17 a^{2} + 22 a + 8\right)\cdot 29^{3} + \left(11 a^{2} + 9 a + 16\right)\cdot 29^{4} + \left(17 a^{2} + 6\right)\cdot 29^{5} + \left(9 a + 1\right)\cdot 29^{6} + \left(15 a^{2} + 8 a + 19\right)\cdot 29^{7} + \left(14 a^{2} + 10 a + 5\right)\cdot 29^{8} + \left(13 a^{2} + 23 a + 5\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 25 a^{2} + 14 a + 23 + \left(6 a^{2} + 7 a + 10\right)\cdot 29 + \left(a^{2} + 13 a + 20\right)\cdot 29^{2} + \left(20 a^{2} + 2 a + 2\right)\cdot 29^{3} + \left(23 a^{2} + a + 13\right)\cdot 29^{4} + \left(22 a^{2} + 23\right)\cdot 29^{5} + \left(16 a^{2} + 22 a + 22\right)\cdot 29^{6} + \left(19 a^{2} + 11 a + 5\right)\cdot 29^{7} + \left(15 a^{2} + 5 a + 7\right)\cdot 29^{8} + \left(11 a^{2} + 21 a + 12\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 28 a^{2} + 12 a + 27 + \left(a + 2\right)\cdot 29 + \left(22 a^{2} + 28 a + 19\right)\cdot 29^{2} + \left(20 a^{2} + 3 a + 3\right)\cdot 29^{3} + \left(22 a^{2} + 18 a + 2\right)\cdot 29^{4} + \left(17 a^{2} + 28 a + 7\right)\cdot 29^{5} + \left(11 a^{2} + 26 a + 6\right)\cdot 29^{6} + \left(23 a^{2} + 8 a + 1\right)\cdot 29^{7} + \left(27 a^{2} + 13 a + 4\right)\cdot 29^{8} + \left(3 a^{2} + 13 a + 2\right)\cdot 29^{9} +O(29^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 14 a^{2} + 18 a + 20 + \left(26 a^{2} + 3 a + 12\right)\cdot 29 + \left(7 a^{2} + 12 a + 19\right)\cdot 29^{2} + \left(13 a^{2} + 13 a + 15\right)\cdot 29^{3} + \left(26 a^{2} + 11\right)\cdot 29^{4} + \left(19 a^{2} + a + 8\right)\cdot 29^{5} + \left(20 a + 28\right)\cdot 29^{6} + \left(3 a^{2} + 8 a + 11\right)\cdot 29^{7} + \left(25 a^{2} + 12 a + 14\right)\cdot 29^{8} + \left(28 a^{2} + 14 a + 5\right)\cdot 29^{9} +O(29^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $12$ |
$9$ | $2$ | $(1,6)$ | $4$ |
$27$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
$27$ | $2$ | $(1,6)(3,8)$ | $0$ |
$6$ | $3$ | $(4,5,7)$ | $0$ |
$8$ | $3$ | $(1,6,9)(2,3,8)(4,5,7)$ | $3$ |
$12$ | $3$ | $(2,3,8)(4,5,7)$ | $-3$ |
$36$ | $3$ | $(1,2,4)(3,5,6)(7,9,8)$ | $0$ |
$36$ | $3$ | $(1,4,2)(3,6,5)(7,8,9)$ | $0$ |
$18$ | $6$ | $(1,6)(4,5,7)$ | $-2$ |
$18$ | $6$ | $(1,6)(2,3,8)$ | $-2$ |
$36$ | $6$ | $(1,6)(2,3,8)(4,5,7)$ | $1$ |
$54$ | $6$ | $(1,6)(3,8)(4,5,7)$ | $0$ |
$108$ | $6$ | $(1,3,5,6,2,4)(7,9,8)$ | $0$ |
$108$ | $6$ | $(1,4,2,6,5,3)(7,8,9)$ | $0$ |
$72$ | $9$ | $(1,2,4,6,3,5,9,8,7)$ | $0$ |
$72$ | $9$ | $(1,4,3,9,7,2,6,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.