Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(224468238494534656\)\(\medspace = 2^{10} \cdot 19^{5} \cdot 97^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.3236131072.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.3236131072.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{3} + 3x + 99 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 81 a^{2} + 33 a + 95 + \left(57 a^{2} + 75 a + 81\right)\cdot 101 + \left(72 a^{2} + 17 a + 77\right)\cdot 101^{2} + \left(17 a^{2} + 96 a + 1\right)\cdot 101^{3} + \left(95 a^{2} + 13 a + 22\right)\cdot 101^{4} + \left(69 a^{2} + 5\right)\cdot 101^{5} + \left(40 a^{2} + 9 a + 14\right)\cdot 101^{6} + \left(34 a^{2} + 16 a + 35\right)\cdot 101^{7} + \left(11 a^{2} + 84 a + 56\right)\cdot 101^{8} + \left(36 a + 67\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 44 a^{2} + 29 a + 21 + \left(28 a^{2} + 82 a + 23\right)\cdot 101 + \left(58 a^{2} + 59 a + 49\right)\cdot 101^{2} + \left(42 a^{2} + 76 a + 51\right)\cdot 101^{3} + \left(79 a^{2} + 58 a + 91\right)\cdot 101^{4} + \left(55 a^{2} + 13 a + 77\right)\cdot 101^{5} + \left(24 a^{2} + 5 a + 82\right)\cdot 101^{6} + \left(24 a^{2} + 23 a + 14\right)\cdot 101^{7} + \left(95 a^{2} + a + 22\right)\cdot 101^{8} + \left(70 a^{2} + 64 a + 7\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 82 + 27\cdot 101 + 31\cdot 101^{2} + 30\cdot 101^{3} + 92\cdot 101^{4} + 85\cdot 101^{5} + 88\cdot 101^{6} + 49\cdot 101^{7} + 49\cdot 101^{8} + 94\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 a^{2} + 21 a + 13 + \left(90 a^{2} + 5 a + 46\right)\cdot 101 + \left(53 a^{2} + 60 a + 40\right)\cdot 101^{2} + \left(28 a^{2} + 56 a + 23\right)\cdot 101^{3} + \left(63 a^{2} + 67 a + 59\right)\cdot 101^{4} + \left(22 a^{2} + 47 a + 11\right)\cdot 101^{5} + \left(65 a^{2} + 84 a + 63\right)\cdot 101^{6} + \left(48 a^{2} + 32 a + 63\right)\cdot 101^{7} + \left(60 a^{2} + 39 a + 53\right)\cdot 101^{8} + \left(20 a^{2} + 55 a + 7\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 42\cdot 101 + 44\cdot 101^{2} + 47\cdot 101^{3} + 83\cdot 101^{4} + 97\cdot 101^{5} + 85\cdot 101^{6} + 65\cdot 101^{7} + 56\cdot 101^{8} + 83\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 a^{2} + 39 a + 87 + \left(14 a^{2} + 44 a + 96\right)\cdot 101 + \left(71 a^{2} + 23 a + 74\right)\cdot 101^{2} + \left(40 a^{2} + 29 a + 47\right)\cdot 101^{3} + \left(27 a^{2} + 28 a + 88\right)\cdot 101^{4} + \left(76 a^{2} + 87 a + 17\right)\cdot 101^{5} + \left(35 a^{2} + 86 a + 4\right)\cdot 101^{6} + \left(42 a^{2} + 61 a + 51\right)\cdot 101^{7} + \left(95 a^{2} + 15 a + 22\right)\cdot 101^{8} + \left(29 a^{2} + 26\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 97 a^{2} + 79 a + 26 + \left(54 a^{2} + 18 a + 76\right)\cdot 101 + \left(58 a^{2} + 17 a + 49\right)\cdot 101^{2} + \left(21 a^{2} + 12 a + 9\right)\cdot 101^{3} + \left(29 a^{2} + 26 a + 92\right)\cdot 101^{4} + \left(58 a^{2} + 29 a + 82\right)\cdot 101^{5} + \left(11 a^{2} + 10 a + 56\right)\cdot 101^{6} + \left(41 a^{2} + 5 a + 48\right)\cdot 101^{7} + \left(57 a^{2} + 20 a + 47\right)\cdot 101^{8} + \left(12 a^{2} + 33 a + 92\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 65 a^{2} + a + 63 + \left(56 a^{2} + 77 a + 79\right)\cdot 101 + \left(89 a^{2} + 23 a + 10\right)\cdot 101^{2} + \left(50 a^{2} + 32 a + 68\right)\cdot 101^{3} + \left(8 a^{2} + 7 a + 50\right)\cdot 101^{4} + \left(20 a^{2} + 24 a + 6\right)\cdot 101^{5} + \left(24 a^{2} + 6 a + 82\right)\cdot 101^{6} + \left(11 a^{2} + 63 a + 89\right)\cdot 101^{7} + \left(84 a^{2} + 41 a + 100\right)\cdot 101^{8} + \left(67 a^{2} + 12 a\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 84 + 30\cdot 101 + 25\cdot 101^{2} + 23\cdot 101^{3} + 26\cdot 101^{4} + 18\cdot 101^{5} + 27\cdot 101^{6} + 86\cdot 101^{7} + 95\cdot 101^{8} + 23\cdot 101^{9} +O(101^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(3,5)$ | $4$ |
| $18$ | $2$ | $(1,3)(2,5)(6,9)$ | $2$ |
| $27$ | $2$ | $(1,2)(3,5)(4,7)$ | $0$ |
| $27$ | $2$ | $(3,5)(4,7)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,4)(5,7)(8,9)$ | $2$ |
| $6$ | $3$ | $(4,7,8)$ | $0$ |
| $8$ | $3$ | $(1,6,2)(3,9,5)(4,8,7)$ | $3$ |
| $12$ | $3$ | $(3,9,5)(4,8,7)$ | $-3$ |
| $72$ | $3$ | $(1,4,3)(2,7,5)(6,8,9)$ | $0$ |
| $54$ | $4$ | $(3,4,5,7)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,2)(3,4,5,7)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,5)(4,7,8)(6,9)$ | $2$ |
| $36$ | $6$ | $(3,4,9,8,5,7)$ | $-1$ |
| $36$ | $6$ | $(3,5)(4,7,8)$ | $-2$ |
| $36$ | $6$ | $(1,2,6)(3,5)(4,7,8)$ | $1$ |
| $54$ | $6$ | $(1,2)(3,5)(4,8,7)$ | $0$ |
| $72$ | $6$ | $(1,9,6,5,2,3)(4,7,8)$ | $-1$ |
| $108$ | $6$ | $(1,2)(3,4,9,8,5,7)$ | $-1$ |
| $216$ | $6$ | $(1,4,5,2,7,3)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,4,9,6,8,5,2,7,3)$ | $0$ |
| $108$ | $12$ | $(1,5,2,3)(4,7,8)(6,9)$ | $0$ |