Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(31826560000\)\(\medspace = 2^{10} \cdot 5^{4} \cdot 223^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.354866144000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.5.354866144000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 4x^{7} - 6x^{6} - x^{5} + 9x^{4} - 21x^{3} + 7x^{2} + 25x + 5 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 20 a^{2} + 62 a + 15 + \left(49 a^{2} + 41 a + 32\right)\cdot 79 + \left(75 a^{2} + 6 a + 32\right)\cdot 79^{2} + \left(68 a^{2} + 35 a + 71\right)\cdot 79^{3} + \left(59 a^{2} + 53 a + 16\right)\cdot 79^{4} + \left(17 a^{2} + 14 a + 1\right)\cdot 79^{5} + \left(11 a^{2} + 51 a + 41\right)\cdot 79^{6} + \left(59 a^{2} + 35 a + 12\right)\cdot 79^{7} + \left(66 a^{2} + 25 a + 58\right)\cdot 79^{8} + \left(48 a^{2} + 43 a + 29\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a^{2} + 40 a + 22 + \left(62 a^{2} + 63 a + 30\right)\cdot 79 + \left(33 a^{2} + 58 a + 18\right)\cdot 79^{2} + \left(67 a^{2} + 54 a + 62\right)\cdot 79^{3} + \left(9 a^{2} + 75 a + 32\right)\cdot 79^{4} + \left(15 a^{2} + 72 a + 64\right)\cdot 79^{5} + \left(52 a^{2} + 77 a + 49\right)\cdot 79^{6} + \left(55 a^{2} + 14 a + 70\right)\cdot 79^{7} + \left(51 a^{2} + 11 a + 46\right)\cdot 79^{8} + \left(78 a^{2} + 36 a + 50\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a^{2} + 22 a + 51 + \left(73 a^{2} + a + 18\right)\cdot 79 + \left(63 a^{2} + 52 a + 41\right)\cdot 79^{2} + \left(25 a^{2} + 36 a + 49\right)\cdot 79^{3} + \left(23 a^{2} + 19 a + 34\right)\cdot 79^{4} + \left(42 a^{2} + 41 a + 69\right)\cdot 79^{5} + \left(5 a^{2} + 3 a + 6\right)\cdot 79^{6} + \left(70 a^{2} + 64 a + 78\right)\cdot 79^{7} + \left(14 a^{2} + 73 a + 62\right)\cdot 79^{8} + \left(37 a^{2} + 23 a + 38\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 51 a^{2} + 56 a + 43 + \left(46 a^{2} + 52 a + 16\right)\cdot 79 + \left(48 a^{2} + 13 a + 28\right)\cdot 79^{2} + \left(21 a^{2} + 68 a + 24\right)\cdot 79^{3} + \left(9 a^{2} + 28 a + 29\right)\cdot 79^{4} + \left(46 a^{2} + 70 a + 13\right)\cdot 79^{5} + \left(15 a^{2} + 28 a + 67\right)\cdot 79^{6} + \left(43 a^{2} + 28 a + 74\right)\cdot 79^{7} + \left(39 a^{2} + 42 a + 52\right)\cdot 79^{8} + \left(30 a^{2} + 78 a + 77\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 69 a^{2} + 52 a + 72 + \left(19 a^{2} + 13\right)\cdot 79 + \left(13 a^{2} + 18 a + 53\right)\cdot 79^{2} + \left(2 a^{2} + 9 a + 65\right)\cdot 79^{3} + \left(21 a^{2} + 72 a + 20\right)\cdot 79^{4} + \left(73 a^{2} + 15 a + 18\right)\cdot 79^{5} + \left(8 a^{2} + 49 a + 27\right)\cdot 79^{6} + \left(69 a^{2} + 43 a + 72\right)\cdot 79^{7} + \left(38 a^{2} + 48\right)\cdot 79^{8} + \left(36 a^{2} + 70 a + 34\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 72 a^{2} + 22 a + 11 + \left(22 a^{2} + 52 a + 32\right)\cdot 79 + \left(75 a^{2} + 77 a + 30\right)\cdot 79^{2} + \left(73 a^{2} + 27 a + 22\right)\cdot 79^{3} + \left(34 a + 58\right)\cdot 79^{4} + \left(11 a^{2} + 31 a + 39\right)\cdot 79^{5} + \left(a^{2} + 21 a + 59\right)\cdot 79^{6} + \left(55 a^{2} + 16 a + 66\right)\cdot 79^{7} + \left(a^{2} + 49 a + 62\right)\cdot 79^{8} + \left(6 a^{2} + 6 a + 9\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 60 a^{2} + 35 a + 18 + \left(61 a^{2} + 25 a + 28\right)\cdot 79 + \left(18 a^{2} + 28 a + 7\right)\cdot 79^{2} + \left(58 a^{2} + 14 a + 7\right)\cdot 79^{3} + \left(54 a^{2} + 25 a + 65\right)\cdot 79^{4} + \left(25 a^{2} + 6 a + 48\right)\cdot 79^{5} + \left(72 a^{2} + 54 a + 12\right)\cdot 79^{6} + \left(32 a^{2} + 77 a + 13\right)\cdot 79^{7} + \left(62 a^{2} + 34 a + 32\right)\cdot 79^{8} + \left(35 a^{2} + 48 a + 30\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 46 a^{2} + 3 a + 13 + \left(75 a^{2} + 6 a + 32\right)\cdot 79 + \left(34 a^{2} + 21 a + 25\right)\cdot 79^{2} + \left(36 a^{2} + 48 a + 34\right)\cdot 79^{3} + \left(24 a^{2} + 69 a + 41\right)\cdot 79^{4} + \left(45 a^{2} + 44 a + 8\right)\cdot 79^{5} + \left(32 a^{2} + 6 a + 11\right)\cdot 79^{6} + \left(75 a^{2} + 73 a + 31\right)\cdot 79^{7} + \left(8 a^{2} + 71 a + 27\right)\cdot 79^{8} + \left(40 a^{2} + 14 a + 56\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 43 a^{2} + 24 a + 74 + \left(62 a^{2} + 72 a + 32\right)\cdot 79 + \left(30 a^{2} + 39 a\right)\cdot 79^{2} + \left(40 a^{2} + 21 a + 58\right)\cdot 79^{3} + \left(33 a^{2} + 16 a + 16\right)\cdot 79^{4} + \left(39 a^{2} + 18 a + 52\right)\cdot 79^{5} + \left(37 a^{2} + 23 a + 40\right)\cdot 79^{6} + \left(13 a^{2} + 41 a + 54\right)\cdot 79^{7} + \left(31 a^{2} + 6 a + 2\right)\cdot 79^{8} + \left(2 a^{2} + 73 a + 67\right)\cdot 79^{9} +O(79^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,3)(2,6)(4,7)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(3,6)(5,8)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,8)(3,6)(4,9)$ | $0$ |
| $6$ | $3$ | $(5,8,9)$ | $-4$ |
| $8$ | $3$ | $(1,2,4)(3,6,7)(5,8,9)$ | $-1$ |
| $12$ | $3$ | $(1,2,4)(5,8,9)$ | $2$ |
| $72$ | $3$ | $(1,3,5)(2,6,8)(4,7,9)$ | $2$ |
| $54$ | $4$ | $(1,3,2,6)(4,7)$ | $0$ |
| $162$ | $4$ | $(1,8,2,5)(3,6)(4,9)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,6)(4,7)(5,8,9)$ | $-2$ |
| $36$ | $6$ | $(1,5,2,8,4,9)$ | $-2$ |
| $36$ | $6$ | $(1,2)(5,8,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,6,7)(5,8,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(3,6)(5,9,8)$ | $0$ |
| $72$ | $6$ | $(1,3,4,7,2,6)(5,8,9)$ | $1$ |
| $108$ | $6$ | $(1,5,2,8,4,9)(3,6)$ | $0$ |
| $216$ | $6$ | $(1,6,8,2,3,5)(4,7,9)$ | $0$ |
| $144$ | $9$ | $(1,3,5,2,6,8,4,7,9)$ | $-1$ |
| $108$ | $12$ | $(1,3,2,6)(4,7)(5,8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.