Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(2854123776\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 7^{2} \cdot 53^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.39217774848.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.39217774848.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{3} + 7x + 59 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 55 a + 30 + \left(37 a^{2} + 17 a + 45\right)\cdot 61 + \left(16 a^{2} + 12 a + 50\right)\cdot 61^{2} + \left(55 a^{2} + 22 a + 21\right)\cdot 61^{3} + \left(12 a^{2} + 39 a + 5\right)\cdot 61^{4} + \left(60 a^{2} + 43 a + 4\right)\cdot 61^{5} + \left(44 a^{2} + 43 a + 55\right)\cdot 61^{6} + \left(4 a^{2} + 53 a + 2\right)\cdot 61^{7} + \left(16 a^{2} + 43 a + 28\right)\cdot 61^{8} + \left(47 a^{2} + 24 a + 46\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 a^{2} + 14 a + 38 + \left(11 a^{2} + 13 a + 59\right)\cdot 61 + \left(22 a^{2} + 51 a + 52\right)\cdot 61^{2} + \left(21 a^{2} + 47 a + 17\right)\cdot 61^{3} + \left(57 a^{2} + 38 a + 24\right)\cdot 61^{4} + \left(2 a^{2} + a + 16\right)\cdot 61^{5} + \left(41 a^{2} + 10 a\right)\cdot 61^{6} + \left(3 a^{2} + 20 a + 26\right)\cdot 61^{7} + \left(48 a^{2} + 29 a + 38\right)\cdot 61^{8} + \left(36 a^{2} + 20 a + 20\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 54 a^{2} + 3 a + 19 + \left(37 a^{2} + 17 a + 39\right)\cdot 61 + \left(56 a^{2} + 3 a + 30\right)\cdot 61^{2} + \left(4 a^{2} + 38 a + 22\right)\cdot 61^{3} + \left(3 a^{2} + 59 a + 35\right)\cdot 61^{4} + \left(39 a^{2} + 56 a + 42\right)\cdot 61^{5} + \left(27 a^{2} + 4 a + 59\right)\cdot 61^{6} + \left(4 a^{2} + 5 a + 49\right)\cdot 61^{7} + \left(2 a^{2} + 23 a + 6\right)\cdot 61^{8} + \left(21 a^{2} + 43 a + 8\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 a^{2} + 19 a + 14 + \left(14 a^{2} + 18 a\right)\cdot 61 + \left(12 a^{2} + 21 a + 10\right)\cdot 61^{2} + \left(12 a^{2} + 55 a + 24\right)\cdot 61^{3} + \left(6 a^{2} + 15 a + 55\right)\cdot 61^{4} + \left(31 a^{2} + 3 a + 30\right)\cdot 61^{5} + \left(33 a^{2} + 57 a + 1\right)\cdot 61^{6} + \left(18 a^{2} + 56 a + 47\right)\cdot 61^{7} + \left(11 a^{2} + 15 a + 5\right)\cdot 61^{8} + \left(37 a^{2} + 6 a + 40\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a^{2} + 30 a + 29 + \left(40 a^{2} + 5 a + 25\right)\cdot 61 + \left(46 a^{2} + 27 a + 10\right)\cdot 61^{2} + \left(32 a^{2} + 41 a + 3\right)\cdot 61^{3} + \left(18 a^{2} + 31 a + 39\right)\cdot 61^{4} + \left(11 a^{2} + 44 a + 1\right)\cdot 61^{5} + \left(25 a^{2} + 54 a + 57\right)\cdot 61^{6} + \left(31 a^{2} + 13 a + 34\right)\cdot 61^{7} + \left(16 a^{2} + 51 a + 45\right)\cdot 61^{8} + \left(20 a^{2} + 33 a + 53\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 46 a^{2} + 39 a + 28 + \left(8 a^{2} + 25 a + 21\right)\cdot 61 + \left(53 a^{2} + 36 a + 20\right)\cdot 61^{2} + \left(43 a^{2} + 28 a + 14\right)\cdot 61^{3} + \left(51 a^{2} + 46 a + 31\right)\cdot 61^{4} + \left(51 a^{2} + 48\right)\cdot 61^{5} + \left(60 a^{2} + 60 a + 60\right)\cdot 61^{6} + \left(37 a^{2} + 59 a + 24\right)\cdot 61^{7} + \left(47 a^{2} + 21 a + 48\right)\cdot 61^{8} + \left(2 a^{2} + 11 a + 12\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 a^{2} + 44 a + 37 + \left(11 a^{2} + 30 a + 37\right)\cdot 61 + \left(43 a^{2} + 6 a + 8\right)\cdot 61^{2} + \left(34 a^{2} + 36 a + 19\right)\cdot 61^{3} + \left(23 a + 3\right)\cdot 61^{4} + \left(19 a^{2} + 2 a + 10\right)\cdot 61^{5} + \left(53 a^{2} + 46 a + 37\right)\cdot 61^{6} + \left(52 a^{2} + 35 a + 11\right)\cdot 61^{7} + \left(10 a^{2} + 8 a + 7\right)\cdot 61^{8} + \left(3 a^{2} + 58 a + 6\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 31 a^{2} + 48 a + 56 + \left(9 a^{2} + 24 a + 58\right)\cdot 61 + \left(32 a^{2} + 27 a + 41\right)\cdot 61^{2} + \left(54 a^{2} + 44 a + 38\right)\cdot 61^{3} + \left(41 a^{2} + 5 a + 18\right)\cdot 61^{4} + \left(30 a^{2} + 14 a + 49\right)\cdot 61^{5} + \left(43 a^{2} + 21 a + 27\right)\cdot 61^{6} + \left(37 a^{2} + 11 a + 14\right)\cdot 61^{7} + \left(33 a^{2} + a + 8\right)\cdot 61^{8} + \left(37 a^{2} + 30 a + 1\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 8 a^{2} + 53 a + 54 + \left(12 a^{2} + 29 a + 16\right)\cdot 61 + \left(22 a^{2} + 58 a + 18\right)\cdot 61^{2} + \left(45 a^{2} + 51 a + 21\right)\cdot 61^{3} + \left(51 a^{2} + 43 a + 31\right)\cdot 61^{4} + \left(58 a^{2} + 15 a + 40\right)\cdot 61^{5} + \left(35 a^{2} + 7 a + 5\right)\cdot 61^{6} + \left(52 a^{2} + 48 a + 32\right)\cdot 61^{7} + \left(57 a^{2} + 48 a + 55\right)\cdot 61^{8} + \left(37 a^{2} + 15 a + 54\right)\cdot 61^{9} +O(61^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,3)(2,4)(6,9)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,4)(5,7)$ | $0$ |
| $27$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $6$ | $3$ | $(5,7,8)$ | $-4$ |
| $8$ | $3$ | $(1,2,9)(3,4,6)(5,7,8)$ | $-1$ |
| $12$ | $3$ | $(3,4,6)(5,7,8)$ | $2$ |
| $72$ | $3$ | $(1,3,5)(2,4,7)(6,8,9)$ | $2$ |
| $54$ | $4$ | $(1,4,2,3)(6,9)$ | $0$ |
| $162$ | $4$ | $(1,7,2,5)(4,6)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,4)(5,7,8)(6,9)$ | $-2$ |
| $36$ | $6$ | $(1,5,2,7,9,8)$ | $-2$ |
| $36$ | $6$ | $(1,2)(5,7,8)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,4,6)(5,7,8)$ | $0$ |
| $54$ | $6$ | $(1,2)(3,4)(5,8,7)$ | $0$ |
| $72$ | $6$ | $(1,3,2,4,9,6)(5,7,8)$ | $1$ |
| $108$ | $6$ | $(1,2)(3,5,4,7,6,8)$ | $0$ |
| $216$ | $6$ | $(1,4,7,2,3,5)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,3,5,2,4,7,9,6,8)$ | $-1$ |
| $108$ | $12$ | $(1,4,2,3)(5,7,8)(6,9)$ | $0$ |