Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(25840385334170624\)\(\medspace = 2^{10} \cdot 7^{4} \cdot 101^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.186476238592.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | even |
| Determinant: | 1.101.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.3.186476238592.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 4x^{7} + 2x^{6} + 2x^{5} + 2x^{4} - 18x^{3} + 28x^{2} - 27x + 9 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{3} + 9x + 76 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + a + 62 + \left(19 a^{2} + 27 a + 77\right)\cdot 79 + \left(6 a^{2} + 70 a + 73\right)\cdot 79^{2} + \left(10 a^{2} + 21 a + 59\right)\cdot 79^{3} + \left(61 a^{2} + a + 76\right)\cdot 79^{4} + \left(27 a^{2} + 58 a + 31\right)\cdot 79^{5} + \left(12 a^{2} + 16 a + 38\right)\cdot 79^{6} + \left(73 a^{2} + 46 a + 39\right)\cdot 79^{7} + \left(49 a^{2} + 73 a + 55\right)\cdot 79^{8} + \left(a^{2} + 57 a + 1\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 77 a^{2} + 49 a + 2 + \left(71 a^{2} + 36 a\right)\cdot 79 + \left(30 a^{2} + 14 a + 64\right)\cdot 79^{2} + \left(33 a^{2} + 33 a + 41\right)\cdot 79^{3} + \left(13 a^{2} + 18 a + 27\right)\cdot 79^{4} + \left(24 a^{2} + 19 a + 10\right)\cdot 79^{5} + \left(7 a^{2} + 36 a + 8\right)\cdot 79^{6} + \left(76 a^{2} + 35 a + 57\right)\cdot 79^{7} + \left(57 a^{2} + 76 a + 24\right)\cdot 79^{8} + \left(74 a^{2} + 45 a + 45\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 21 a^{2} + 52 a + 46 + \left(32 a^{2} + 22 a + 39\right)\cdot 79 + \left(35 a^{2} + 45 a + 23\right)\cdot 79^{2} + \left(71 a^{2} + 13 a + 3\right)\cdot 79^{3} + \left(35 a^{2} + 67 a + 10\right)\cdot 79^{4} + \left(8 a^{2} + 24 a + 28\right)\cdot 79^{5} + \left(37 a^{2} + 72 a + 76\right)\cdot 79^{6} + \left(41 a^{2} + 14 a + 30\right)\cdot 79^{7} + \left(32 a^{2} + 19 a + 1\right)\cdot 79^{8} + \left(28 a^{2} + 19 a + 59\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 4 a^{2} + 7 a + 23 + \left(2 a^{2} + 29 a + 16\right)\cdot 79 + \left(76 a^{2} + 5 a + 30\right)\cdot 79^{2} + \left(21 a^{2} + 50 a + 22\right)\cdot 79^{3} + \left(66 a^{2} + 22 a + 34\right)\cdot 79^{4} + \left(38 a^{2} + 76 a + 52\right)\cdot 79^{5} + \left(34 a^{2} + 53 a + 60\right)\cdot 79^{6} + \left(77 a^{2} + 56 a + 9\right)\cdot 79^{7} + \left(69 a^{2} + 66 a + 68\right)\cdot 79^{8} + \left(69 a^{2} + 17 a + 70\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 54 a^{2} + 20 a + 7 + \left(44 a^{2} + 27 a + 35\right)\cdot 79 + \left(46 a^{2} + 28 a + 11\right)\cdot 79^{2} + \left(64 a^{2} + 15 a + 41\right)\cdot 79^{3} + \left(55 a^{2} + 68 a + 50\right)\cdot 79^{4} + \left(31 a^{2} + 56 a + 9\right)\cdot 79^{5} + \left(7 a^{2} + 31 a + 56\right)\cdot 79^{6} + \left(39 a^{2} + 7 a + 16\right)\cdot 79^{7} + \left(55 a^{2} + 72 a + 60\right)\cdot 79^{8} + \left(59 a^{2} + 41 a + 9\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 13 a^{2} + 61 a + 39 + \left(76 a^{2} + 29 a + 68\right)\cdot 79 + \left(52 a^{2} + 47 a + 48\right)\cdot 79^{2} + \left(6 a^{2} + 42 a + 44\right)\cdot 79^{3} + \left(68 a^{2} + 17 a + 8\right)\cdot 79^{4} + \left(45 a^{2} + 28 a + 11\right)\cdot 79^{5} + \left(10 a^{2} + 53 a + 61\right)\cdot 79^{6} + \left(69 a^{2} + 19 a + 57\right)\cdot 79^{7} + \left(67 a^{2} + 46 a + 28\right)\cdot 79^{8} + \left(67 a^{2} + 46 a + 26\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 73 a^{2} + 29 a + 57 + \left(66 a^{2} + 15 a + 48\right)\cdot 79 + \left(41 a^{2} + 73 a + 50\right)\cdot 79^{2} + \left(35 a^{2} + 23 a + 54\right)\cdot 79^{3} + \left(4 a^{2} + 59 a + 52\right)\cdot 79^{4} + \left(27 a^{2} + a + 27\right)\cdot 79^{5} + \left(59 a^{2} + 26 a + 4\right)\cdot 79^{6} + \left(8 a^{2} + 76 a + 48\right)\cdot 79^{7} + \left(50 a^{2} + 7 a + 56\right)\cdot 79^{8} + \left(2 a^{2} + 54 a + 7\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 43 a^{2} + 29 a + 61 + \left(15 a^{2} + 45 a + 20\right)\cdot 79 + \left(75 a^{2} + 24 a + 24\right)\cdot 79^{2} + \left(67 a^{2} + 73 a + 17\right)\cdot 79^{3} + \left(19 a^{2} + 54 a + 35\right)\cdot 79^{4} + \left(65 a^{2} + 70 a + 48\right)\cdot 79^{5} + \left(58 a^{2} + 77 a + 34\right)\cdot 79^{6} + \left(a^{2} + 25 a + 48\right)\cdot 79^{7} + \left(65 a^{2} + 31 a + 11\right)\cdot 79^{8} + \left(30 a^{2} + 41\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 23 a^{2} + 68 a + 20 + \left(66 a^{2} + 3 a + 9\right)\cdot 79 + \left(29 a^{2} + 7 a + 68\right)\cdot 79^{2} + \left(4 a^{2} + 42 a + 30\right)\cdot 79^{3} + \left(70 a^{2} + 6 a + 20\right)\cdot 79^{4} + \left(46 a^{2} + 59 a + 17\right)\cdot 79^{5} + \left(9 a^{2} + 26 a + 55\right)\cdot 79^{6} + \left(8 a^{2} + 33 a + 7\right)\cdot 79^{7} + \left(25 a^{2} + a + 9\right)\cdot 79^{8} + \left(59 a^{2} + 32 a + 54\right)\cdot 79^{9} +O(79^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(1,4)$ | $4$ |
| $18$ | $2$ | $(1,5)(4,7)(6,8)$ | $2$ |
| $27$ | $2$ | $(1,4)(2,3)(5,7)$ | $0$ |
| $27$ | $2$ | $(1,4)(5,7)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,5)(3,7)(8,9)$ | $2$ |
| $6$ | $3$ | $(2,3,9)$ | $0$ |
| $8$ | $3$ | $(1,4,6)(2,3,9)(5,7,8)$ | $3$ |
| $12$ | $3$ | $(1,4,6)(2,3,9)$ | $-3$ |
| $72$ | $3$ | $(1,5,2)(3,4,7)(6,8,9)$ | $0$ |
| $54$ | $4$ | $(1,7,4,5)(6,8)$ | $0$ |
| $162$ | $4$ | $(1,3,4,2)(6,9)(7,8)$ | $0$ |
| $36$ | $6$ | $(1,5)(2,3,9)(4,7)(6,8)$ | $2$ |
| $36$ | $6$ | $(1,2,4,3,6,9)$ | $-1$ |
| $36$ | $6$ | $(1,4)(2,3,9)$ | $-2$ |
| $36$ | $6$ | $(1,4)(2,3,9)(5,7,8)$ | $1$ |
| $54$ | $6$ | $(1,4)(2,9,3)(5,7)$ | $0$ |
| $72$ | $6$ | $(1,5,4,7,6,8)(2,3,9)$ | $-1$ |
| $108$ | $6$ | $(1,4)(2,7,3,8,9,5)$ | $-1$ |
| $216$ | $6$ | $(1,7,3,4,5,2)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,5,2,4,7,3,6,8,9)$ | $0$ |
| $108$ | $12$ | $(1,7,4,5)(2,3,9)(6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.