Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(218768667202582641\)\(\medspace = 3^{20} \cdot 89^{4}\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.3831158169.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_3^3:C_2^2.C_6$ |
| Projective stem field: | Galois closure of 9.5.3831158169.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 3x^{7} - 6x^{5} + 6x^{4} - 3x^{3} - 3x^{2} + 3x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{3} + 3x + 51 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 43 a^{2} + 32 a + 45 + \left(38 a^{2} + 49 a + 52\right)\cdot 53 + \left(20 a^{2} + 40 a + 1\right)\cdot 53^{2} + \left(14 a^{2} + 5 a + 11\right)\cdot 53^{3} + \left(8 a^{2} + 14 a + 44\right)\cdot 53^{4} + \left(4 a^{2} + 26 a + 25\right)\cdot 53^{5} + \left(49 a^{2} + 23 a + 12\right)\cdot 53^{6} + \left(8 a^{2} + 50 a + 23\right)\cdot 53^{7} + \left(50 a^{2} + 9 a + 19\right)\cdot 53^{8} + \left(42 a^{2} + 16 a\right)\cdot 53^{9} +O(53^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 26 a^{2} + 48 a + 4 + \left(20 a^{2} + 38 a + 6\right)\cdot 53 + \left(38 a^{2} + 19 a + 50\right)\cdot 53^{2} + \left(36 a^{2} + 26 a + 5\right)\cdot 53^{3} + \left(34 a^{2} + 23 a + 50\right)\cdot 53^{4} + \left(16 a^{2} + 50 a + 40\right)\cdot 53^{5} + \left(28 a^{2} + 3 a + 12\right)\cdot 53^{6} + \left(44 a^{2} + 28 a + 50\right)\cdot 53^{7} + \left(17 a^{2} + 26 a + 20\right)\cdot 53^{8} + \left(42 a^{2} + 39 a + 20\right)\cdot 53^{9} +O(53^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 4 a^{2} + 39 a + 20 + \left(52 a^{2} + 15 a + 26\right)\cdot 53 + \left(29 a^{2} + 51 a + 20\right)\cdot 53^{2} + \left(44 a^{2} + 50 a + 18\right)\cdot 53^{3} + \left(3 a^{2} + 18 a + 35\right)\cdot 53^{4} + \left(12 a^{2} + 10 a + 41\right)\cdot 53^{5} + \left(38 a^{2} + 23 a + 43\right)\cdot 53^{6} + \left(39 a^{2} + 41 a + 31\right)\cdot 53^{7} + \left(16 a^{2} + 34 a + 5\right)\cdot 53^{8} + \left(43 a^{2} + 13 a + 1\right)\cdot 53^{9} +O(53^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 30 + 7\cdot 53 + 47\cdot 53^{2} + 35\cdot 53^{3} + 12\cdot 53^{4} + 43\cdot 53^{5} + 35\cdot 53^{6} + 5\cdot 53^{7} + 3\cdot 53^{8} + 32\cdot 53^{9} +O(53^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 38 a^{2} + 3 a + 28 + \left(51 a^{2} + 49 a + 15\right)\cdot 53 + \left(22 a^{2} + 15 a + 19\right)\cdot 53^{2} + \left(52 a^{2} + 27 a + 37\right)\cdot 53^{3} + \left(34 a^{2} + 43 a + 50\right)\cdot 53^{4} + \left(2 a^{2} + 4 a + 12\right)\cdot 53^{5} + \left(24 a^{2} + 30 a + 4\right)\cdot 53^{6} + \left(46 a^{2} + 13 a + 1\right)\cdot 53^{7} + \left(44 a^{2} + 21 a + 22\right)\cdot 53^{8} + \left(27 a^{2} + 5 a + 44\right)\cdot 53^{9} +O(53^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 42 a^{2} + 2 a + 36 + \left(33 a^{2} + 18 a + 32\right)\cdot 53 + \left(44 a^{2} + 17 a + 9\right)\cdot 53^{2} + \left(16 a^{2} + 52 a + 19\right)\cdot 53^{3} + \left(36 a^{2} + 38 a\right)\cdot 53^{4} + \left(33 a^{2} + 50 a + 22\right)\cdot 53^{5} + \left(18 a + 10\right)\cdot 53^{6} + \left(15 a^{2} + 11 a + 44\right)\cdot 53^{7} + \left(43 a^{2} + 5 a + 18\right)\cdot 53^{8} + \left(35 a^{2} + 8 a + 7\right)\cdot 53^{9} +O(53^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 31 + 24\cdot 53 + 50\cdot 53^{2} + 7\cdot 53^{3} + 39\cdot 53^{4} + 44\cdot 53^{5} + 7\cdot 53^{6} + 7\cdot 53^{7} + 17\cdot 53^{8} + 15\cdot 53^{9} +O(53^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 50 + 40\cdot 53 + 47\cdot 53^{2} + 52\cdot 53^{3} + 28\cdot 53^{4} + 48\cdot 53^{5} + 26\cdot 53^{6} + 34\cdot 53^{7} + 53^{8} + 31\cdot 53^{9} +O(53^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 6 a^{2} + 35 a + 24 + \left(15 a^{2} + 40 a + 5\right)\cdot 53 + \left(2 a^{2} + 13 a + 18\right)\cdot 53^{2} + \left(47 a^{2} + 49 a + 23\right)\cdot 53^{3} + \left(40 a^{2} + 19 a + 3\right)\cdot 53^{4} + \left(36 a^{2} + 16 a + 38\right)\cdot 53^{5} + \left(18 a^{2} + 6 a + 4\right)\cdot 53^{6} + \left(4 a^{2} + 14 a + 14\right)\cdot 53^{7} + \left(39 a^{2} + 8 a + 50\right)\cdot 53^{8} + \left(19 a^{2} + 23 a + 6\right)\cdot 53^{9} +O(53^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(1,3)$ | $4$ |
| $27$ | $2$ | $(1,3)(2,5)(4,7)$ | $0$ |
| $27$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $6$ | $3$ | $(4,7,8)$ | $0$ |
| $8$ | $3$ | $(1,3,9)(2,5,6)(4,7,8)$ | $3$ |
| $12$ | $3$ | $(2,5,6)(4,7,8)$ | $-3$ |
| $36$ | $3$ | $(1,4,2)(3,7,5)(6,9,8)$ | $0$ |
| $36$ | $3$ | $(1,2,4)(3,5,7)(6,8,9)$ | $0$ |
| $18$ | $6$ | $(1,3)(4,7,8)$ | $-2$ |
| $18$ | $6$ | $(1,3)(2,5,6)$ | $-2$ |
| $36$ | $6$ | $(1,3)(2,5,6)(4,7,8)$ | $1$ |
| $54$ | $6$ | $(1,3)(4,7,8)(5,6)$ | $0$ |
| $108$ | $6$ | $(1,7,5,3,4,2)(6,9,8)$ | $0$ |
| $108$ | $6$ | $(1,2,4,3,5,7)(6,8,9)$ | $0$ |
| $72$ | $9$ | $(1,4,5,3,7,6,9,8,2)$ | $0$ |
| $72$ | $9$ | $(1,5,7,9,2,4,3,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.