Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(37636000000\)\(\medspace = 2^{8} \cdot 5^{6} \cdot 97^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.146027680000.1 |
| 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 9.5.146027680000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 8x^{6} - 6x^{5} - 6x^{4} + 8x^{2} + 5x - 5 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 31 a^{2} + 49 a + 58 + \left(20 a^{2} + 67 a + 71\right)\cdot 79 + \left(73 a^{2} + 51 a + 57\right)\cdot 79^{2} + \left(58 a^{2} + 18 a + 36\right)\cdot 79^{3} + \left(19 a^{2} + 23 a + 29\right)\cdot 79^{4} + \left(5 a^{2} + 43 a + 47\right)\cdot 79^{5} + \left(64 a^{2} + 16 a + 59\right)\cdot 79^{6} + \left(31 a^{2} + 13 a + 46\right)\cdot 79^{7} + \left(14 a^{2} + 65 a + 24\right)\cdot 79^{8} + \left(45 a^{2} + 29 a + 35\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a^{2} + 30 a + 44 + \left(14 a^{2} + 73 a + 72\right)\cdot 79 + \left(37 a^{2} + 7 a + 2\right)\cdot 79^{2} + \left(40 a^{2} + 20 a + 5\right)\cdot 79^{3} + \left(8 a^{2} + 31 a + 54\right)\cdot 79^{4} + \left(32 a^{2} + 30 a + 1\right)\cdot 79^{5} + \left(33 a^{2} + 78 a + 76\right)\cdot 79^{6} + \left(49 a^{2} + 33 a + 39\right)\cdot 79^{7} + \left(7 a^{2} + 6 a + 52\right)\cdot 79^{8} + \left(9 a^{2} + 17 a + 59\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 28 a^{2} + 42 a + 40 + \left(70 a^{2} + 43 a + 55\right)\cdot 79 + \left(67 a^{2} + 24 a + 25\right)\cdot 79^{2} + \left(13 a^{2} + 45 a + 3\right)\cdot 79^{3} + \left(33 a^{2} + 77 a + 31\right)\cdot 79^{4} + \left(28 a^{2} + 53 a + 28\right)\cdot 79^{5} + \left(52 a^{2} + 57 a + 68\right)\cdot 79^{6} + \left(2 a^{2} + 38 a + 29\right)\cdot 79^{7} + \left(75 a^{2} + 51 a + 72\right)\cdot 79^{8} + \left(28 a^{2} + 7 a + 16\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 a^{2} + 22 a + 49 + \left(45 a + 68\right)\cdot 79 + \left(19 a^{2} + 4 a + 51\right)\cdot 79^{2} + \left(49 a^{2} + 59 a + 57\right)\cdot 79^{3} + \left(15 a^{2} + 55 a + 17\right)\cdot 79^{4} + \left(33 a^{2} + 21 a + 8\right)\cdot 79^{5} + \left(5 a^{2} + 42 a + 66\right)\cdot 79^{6} + \left(67 a^{2} + 67 a + 66\right)\cdot 79^{7} + \left(32 a^{2} + a + 45\right)\cdot 79^{8} + \left(63 a^{2} + 27 a + 69\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 a^{2} + 15 a + 70 + \left(8 a^{2} + 69 a + 33\right)\cdot 79 + \left(71 a^{2} + 49 a + 1\right)\cdot 79^{2} + \left(15 a^{2} + 53 a + 18\right)\cdot 79^{3} + \left(30 a^{2} + 24 a + 30\right)\cdot 79^{4} + \left(17 a^{2} + 3 a + 42\right)\cdot 79^{5} + \left(21 a^{2} + 58 a + 23\right)\cdot 79^{6} + \left(9 a^{2} + 51 a + 61\right)\cdot 79^{7} + \left(50 a^{2} + 25 a + 39\right)\cdot 79^{8} + \left(65 a^{2} + 44 a + 71\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a^{2} + 64 a + 21 + \left(26 a^{2} + 71 a + 63\right)\cdot 79 + \left(39 a^{2} + 9 a + 47\right)\cdot 79^{2} + \left(4 a^{2} + 64 a + 28\right)\cdot 79^{3} + \left(77 a^{2} + 29 a + 74\right)\cdot 79^{4} + \left(19 a^{2} + 70 a + 57\right)\cdot 79^{5} + \left(76 a^{2} + 36 a + 37\right)\cdot 79^{6} + \left(71 a^{2} + 74 a + 42\right)\cdot 79^{7} + \left(50 a^{2} + 45 a + 44\right)\cdot 79^{8} + \left(67 a^{2} + 2 a + 4\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 a^{2} + 57 + \left(44 a^{2} + 17 a + 13\right)\cdot 79 + \left(47 a^{2} + 19 a + 18\right)\cdot 79^{2} + \left(58 a^{2} + 40 a + 37\right)\cdot 79^{3} + \left(50 a^{2} + 24 a + 74\right)\cdot 79^{4} + \left(41 a^{2} + 5 a + 29\right)\cdot 79^{5} + \left(60 a^{2} + 63 a + 22\right)\cdot 79^{6} + \left(76 a^{2} + 31 a + 71\right)\cdot 79^{7} + \left(56 a^{2} + 7 a + 1\right)\cdot 79^{8} + \left(24 a^{2} + 32 a + 63\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 20 a^{2} + 67 a + 71 + \left(67 a^{2} + 46 a + 36\right)\cdot 79 + \left(16 a^{2} + 2 a + 35\right)\cdot 79^{2} + \left(6 a^{2} + 15 a + 36\right)\cdot 79^{3} + \left(26 a^{2} + 57 a + 67\right)\cdot 79^{4} + \left(45 a^{2} + 60 a + 50\right)\cdot 79^{5} + \left(41 a^{2} + 4 a + 3\right)\cdot 79^{6} + \left(44 a^{2} + 27 a + 44\right)\cdot 79^{7} + \left(68 a^{2} + 41 a + 33\right)\cdot 79^{8} + \left(4 a^{2} + 41 a + 30\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 33 a^{2} + 27 a + 67 + \left(64 a^{2} + 39 a + 57\right)\cdot 79 + \left(22 a^{2} + 66 a + 74\right)\cdot 79^{2} + \left(68 a^{2} + 78 a + 13\right)\cdot 79^{3} + \left(54 a^{2} + 70 a + 16\right)\cdot 79^{4} + \left(13 a^{2} + 26 a + 49\right)\cdot 79^{5} + \left(40 a^{2} + 37 a + 37\right)\cdot 79^{6} + \left(41 a^{2} + 56 a + 71\right)\cdot 79^{7} + \left(38 a^{2} + 70 a\right)\cdot 79^{8} + \left(6 a^{2} + 34 a + 44\right)\cdot 79^{9} +O(79^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(6,8)$ | $0$ |
| $18$ | $2$ | $(1,6)(2,8)(7,9)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,4)(6,8)$ | $0$ |
| $27$ | $2$ | $(1,2)(6,8)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $6$ | $3$ | $(3,4,5)$ | $-4$ |
| $8$ | $3$ | $(1,2,7)(3,4,5)(6,8,9)$ | $-1$ |
| $12$ | $3$ | $(1,2,7)(3,4,5)$ | $2$ |
| $72$ | $3$ | $(1,3,6)(2,4,8)(5,9,7)$ | $2$ |
| $54$ | $4$ | $(1,6,2,8)(7,9)$ | $0$ |
| $162$ | $4$ | $(2,7)(3,6,4,8)(5,9)$ | $0$ |
| $36$ | $6$ | $(1,6)(2,8)(3,4,5)(7,9)$ | $-2$ |
| $36$ | $6$ | $(3,8,4,9,5,6)$ | $-2$ |
| $36$ | $6$ | $(3,4,5)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,2,7)(3,4,5)(6,8)$ | $0$ |
| $54$ | $6$ | $(1,2)(3,5,4)(6,8)$ | $0$ |
| $72$ | $6$ | $(1,8,2,9,7,6)(3,4,5)$ | $1$ |
| $108$ | $6$ | $(1,3,2,4,7,5)(6,8)$ | $0$ |
| $216$ | $6$ | $(1,3,6,2,4,8)(5,9,7)$ | $0$ |
| $144$ | $9$ | $(1,3,8,2,4,9,7,5,6)$ | $-1$ |
| $108$ | $12$ | $(1,6,2,8)(3,4,5)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.