Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(14113440000\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 5^{4} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.431244000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.431244000000.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{3} + 9x + 76 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 74 a^{2} + 72 a + 67 + \left(35 a^{2} + 64 a + 27\right)\cdot 79 + \left(45 a^{2} + 9 a + 70\right)\cdot 79^{2} + \left(57 a^{2} + 35 a + 64\right)\cdot 79^{3} + \left(18 a^{2} + 27 a + 55\right)\cdot 79^{4} + \left(45 a^{2} + 57 a + 37\right)\cdot 79^{5} + \left(11 a^{2} + 51 a + 15\right)\cdot 79^{6} + \left(50 a^{2} + 78 a + 40\right)\cdot 79^{7} + \left(32 a^{2} + 28 a + 49\right)\cdot 79^{8} + \left(58 a^{2} + 47 a + 38\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 58 a^{2} + 51 a + 46 + \left(8 a^{2} + 26\right)\cdot 79 + \left(24 a^{2} + 69 a + 30\right)\cdot 79^{2} + \left(78 a^{2} + 34 a + 15\right)\cdot 79^{3} + \left(25 a^{2} + 3 a + 43\right)\cdot 79^{4} + \left(33 a^{2} + 20 a + 31\right)\cdot 79^{5} + \left(30 a^{2} + 55 a + 34\right)\cdot 79^{6} + \left(35 a^{2} + 68 a + 18\right)\cdot 79^{7} + \left(31 a^{2} + 66 a + 46\right)\cdot 79^{8} + \left(44 a^{2} + 49 a + 15\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a^{2} + 39 a + 68 + \left(45 a^{2} + 2 a + 7\right)\cdot 79 + \left(7 a^{2} + 75 a + 10\right)\cdot 79^{2} + \left(57 a^{2} + 29 a + 46\right)\cdot 79^{3} + \left(77 a^{2} + 73 a + 37\right)\cdot 79^{4} + \left(5 a^{2} + 63 a + 25\right)\cdot 79^{5} + \left(5 a^{2} + 69 a + 40\right)\cdot 79^{6} + \left(57 a^{2} + 38 a + 69\right)\cdot 79^{7} + \left(55 a^{2} + 74 a + 33\right)\cdot 79^{8} + \left(41 a^{2} + 51 a + 78\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 72 a^{2} + 18 a + 6 + \left(17 a^{2} + 17 a + 5\right)\cdot 79 + \left(65 a^{2} + 55 a + 76\right)\cdot 79^{2} + \left(77 a^{2} + 29 a + 16\right)\cdot 79^{3} + \left(5 a^{2} + 49 a + 47\right)\cdot 79^{4} + \left(73 a^{2} + 26 a + 50\right)\cdot 79^{5} + \left(23 a^{2} + 73 a + 29\right)\cdot 79^{6} + \left(42 a^{2} + 28 a + 76\right)\cdot 79^{7} + \left(54 a^{2} + 33 a + 62\right)\cdot 79^{8} + \left(27 a^{2} + 54 a + 17\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a^{2} + 68 a + 7 + \left(25 a^{2} + 75 a + 46\right)\cdot 79 + \left(47 a^{2} + 13 a + 11\right)\cdot 79^{2} + \left(22 a^{2} + 14 a + 76\right)\cdot 79^{3} + \left(54 a^{2} + 2 a + 54\right)\cdot 79^{4} + \left(39 a^{2} + 74 a + 69\right)\cdot 79^{5} + \left(43 a^{2} + 32 a + 33\right)\cdot 79^{6} + \left(65 a^{2} + 50 a + 41\right)\cdot 79^{7} + \left(70 a^{2} + 16 a + 45\right)\cdot 79^{8} + \left(71 a^{2} + 56 a + 22\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 39 a^{2} + 57 a + 15 + \left(27 a^{2} + 48 a + 56\right)\cdot 79 + \left(6 a^{2} + 74 a + 72\right)\cdot 79^{2} + \left(32 a^{2} + 24 a + 69\right)\cdot 79^{3} + \left(33 a^{2} + 51 a + 64\right)\cdot 79^{4} + \left(78 a^{2} + 34 a + 78\right)\cdot 79^{5} + \left(70 a^{2} + 15 a + 55\right)\cdot 79^{6} + \left(62 a^{2} + 18 a + 37\right)\cdot 79^{7} + \left(14 a^{2} + 39 a + 21\right)\cdot 79^{8} + \left(73 a^{2} + 26 a + 48\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 45 a^{2} + 29 a + 51 + \left(15 a^{2} + 44 a + 63\right)\cdot 79 + \left(27 a^{2} + 73 a + 39\right)\cdot 79^{2} + \left(68 a^{2} + 18 a + 50\right)\cdot 79^{3} + \left(26 a^{2} + 25\right)\cdot 79^{4} + \left(34 a^{2} + 66 a + 51\right)\cdot 79^{5} + \left(75 a^{2} + 11 a + 3\right)\cdot 79^{6} + \left(44 a^{2} + 61 a + 9\right)\cdot 79^{7} + \left(31 a^{2} + 10 a + 43\right)\cdot 79^{8} + \left(26 a^{2} + 5 a + 4\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 31 a^{2} + 62 a + 76 + \left(6 a^{2} + 27 a + 14\right)\cdot 79 + \left(65 a^{2} + 8 a + 75\right)\cdot 79^{2} + \left(68 a^{2} + 24 a + 41\right)\cdot 79^{3} + \left(46 a^{2} + 33 a + 55\right)\cdot 79^{4} + \left(73 a^{2} + 59 a + 53\right)\cdot 79^{5} + \left(2 a^{2} + 72 a + 61\right)\cdot 79^{6} + \left(38 a^{2} + 21 a + 50\right)\cdot 79^{7} + \left(8 a^{2} + 44 a + 23\right)\cdot 79^{8} + \left(43 a^{2} + 31\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 55 a^{2} + 78 a + 62 + \left(54 a^{2} + 33 a + 67\right)\cdot 79 + \left(27 a^{2} + 15 a + 8\right)\cdot 79^{2} + \left(11 a^{2} + 25 a + 13\right)\cdot 79^{3} + \left(26 a^{2} + 75 a + 10\right)\cdot 79^{4} + \left(11 a^{2} + 71 a + 75\right)\cdot 79^{5} + \left(52 a^{2} + 11 a + 40\right)\cdot 79^{6} + \left(77 a^{2} + 28 a + 51\right)\cdot 79^{7} + \left(15 a^{2} + a + 68\right)\cdot 79^{8} + \left(8 a^{2} + 24 a + 58\right)\cdot 79^{9} +O(79^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,4)$ | $0$ |
| $18$ | $2$ | $(1,3)(4,6)(5,8)$ | $4$ |
| $27$ | $2$ | $(1,4)(2,7)(3,6)$ | $0$ |
| $27$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,3)(6,7)(8,9)$ | $0$ |
| $6$ | $3$ | $(2,7,9)$ | $-4$ |
| $8$ | $3$ | $(1,4,5)(2,7,9)(3,6,8)$ | $-1$ |
| $12$ | $3$ | $(2,7,9)(3,6,8)$ | $2$ |
| $72$ | $3$ | $(1,3,2)(4,6,7)(5,8,9)$ | $2$ |
| $54$ | $4$ | $(1,6,4,3)(5,8)$ | $0$ |
| $162$ | $4$ | $(1,7,4,2)(5,9)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,7,9)(4,6)(5,8)$ | $-2$ |
| $36$ | $6$ | $(1,2,4,7,5,9)$ | $-2$ |
| $36$ | $6$ | $(1,4)(2,7,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,7,9)(3,6,8)$ | $0$ |
| $54$ | $6$ | $(1,4)(2,9,7)(3,6)$ | $0$ |
| $72$ | $6$ | $(1,3,4,6,5,8)(2,7,9)$ | $1$ |
| $108$ | $6$ | $(1,4)(2,6,7,8,9,3)$ | $0$ |
| $216$ | $6$ | $(1,6,7,4,3,2)(5,8,9)$ | $0$ |
| $144$ | $9$ | $(1,3,2,4,6,7,5,8,9)$ | $-1$ |
| $108$ | $12$ | $(1,6,4,3)(2,7,9)(5,8)$ | $0$ |