Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(53995089429705152\)\(\medspace = 2^{6} \cdot 23^{5} \cdot 107^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.83319616.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.83319616.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 45 a^{2} + 93 a + 92 + \left(99 a^{2} + 54 a + 25\right)\cdot 101 + \left(2 a^{2} + 78 a + 2\right)\cdot 101^{2} + \left(3 a^{2} + 44 a + 2\right)\cdot 101^{3} + \left(64 a^{2} + 96 a + 47\right)\cdot 101^{4} + \left(66 a^{2} + 56 a + 2\right)\cdot 101^{5} + \left(67 a^{2} + 13 a + 62\right)\cdot 101^{6} + \left(63 a^{2} + 50 a + 60\right)\cdot 101^{7} + \left(a^{2} + 82 a + 59\right)\cdot 101^{8} + \left(63 a^{2} + 33 a + 71\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 42 a^{2} + 68 a + 97 + \left(8 a^{2} + 92 a + 1\right)\cdot 101 + \left(74 a^{2} + 47 a + 93\right)\cdot 101^{2} + \left(36 a^{2} + 29 a + 13\right)\cdot 101^{3} + \left(72 a^{2} + 94 a + 38\right)\cdot 101^{4} + \left(88 a^{2} + 16 a + 39\right)\cdot 101^{5} + \left(38 a^{2} + 14 a + 71\right)\cdot 101^{6} + \left(86 a^{2} + 69 a + 54\right)\cdot 101^{7} + \left(90 a^{2} + 83 a + 27\right)\cdot 101^{8} + \left(25 a^{2} + 47 a + 78\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 90 a^{2} + 86 a + 92 + \left(a^{2} + 71 a + 89\right)\cdot 101 + \left(20 a^{2} + 22 a + 85\right)\cdot 101^{2} + \left(36 a^{2} + 34 a + 12\right)\cdot 101^{3} + \left(58 a^{2} + 50 a + 10\right)\cdot 101^{4} + \left(26 a^{2} + 19 a + 16\right)\cdot 101^{5} + \left(37 a^{2} + 96 a + 68\right)\cdot 101^{6} + \left(88 a^{2} + 45 a + 58\right)\cdot 101^{7} + \left(3 a^{2} + 18 a + 55\right)\cdot 101^{8} + \left(99 a^{2} + 40 a + 22\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 57 + 52\cdot 101 + 18\cdot 101^{2} + 15\cdot 101^{3} + 43\cdot 101^{4} + 46\cdot 101^{5} + 37\cdot 101^{6} + 36\cdot 101^{7} + 9\cdot 101^{8} + 16\cdot 101^{9} +O(101^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 42 + 90\cdot 101 + 99\cdot 101^{2} + 84\cdot 101^{3} + 86\cdot 101^{4} + 63\cdot 101^{5} + 53\cdot 101^{6} + 60\cdot 101^{7} + 69\cdot 101^{8} + 81\cdot 101^{9} +O(101^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 70 a^{2} + 48 a + 52 + \left(90 a^{2} + 37 a + 65\right)\cdot 101 + \left(6 a^{2} + 30 a + 59\right)\cdot 101^{2} + \left(28 a^{2} + 37 a + 97\right)\cdot 101^{3} + \left(71 a^{2} + 57 a + 35\right)\cdot 101^{4} + \left(86 a^{2} + 64 a + 35\right)\cdot 101^{5} + \left(24 a^{2} + 91 a + 43\right)\cdot 101^{6} + \left(27 a^{2} + 86 a + 37\right)\cdot 101^{7} + \left(6 a^{2} + 99 a + 60\right)\cdot 101^{8} + \left(77 a^{2} + 12 a + 79\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 62 + 16\cdot 101 + 58\cdot 101^{2} + 90\cdot 101^{3} + 28\cdot 101^{4} + 90\cdot 101^{5} + 46\cdot 101^{6} + 53\cdot 101^{7} + 12\cdot 101^{8} + 87\cdot 101^{9} +O(101^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 62 a^{2} + 26 a + 25 + \left(83 a^{2} + 50 a + 95\right)\cdot 101 + \left(96 a^{2} + 16 a + 88\right)\cdot 101^{2} + \left(50 a^{2} + 80 a + 97\right)\cdot 101^{3} + \left(91 a^{2} + 97 a\right)\cdot 101^{4} + \left(63 a^{2} + 87 a + 98\right)\cdot 101^{5} + \left(25 a^{2} + 75 a + 78\right)\cdot 101^{6} + \left(94 a^{2} + 9 a + 20\right)\cdot 101^{7} + \left(70 a^{2} + 15 a + 97\right)\cdot 101^{8} + \left(40 a^{2} + 8 a + 26\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 95 a^{2} + 83 a + 91 + \left(18 a^{2} + 96 a + 66\right)\cdot 101 + \left(a^{2} + 5 a + 99\right)\cdot 101^{2} + \left(47 a^{2} + 77 a + 89\right)\cdot 101^{3} + \left(46 a^{2} + 7 a + 11\right)\cdot 101^{4} + \left(71 a^{2} + 57 a + 12\right)\cdot 101^{5} + \left(7 a^{2} + 11 a + 43\right)\cdot 101^{6} + \left(44 a^{2} + 41 a + 21\right)\cdot 101^{7} + \left(28 a^{2} + 3 a + 12\right)\cdot 101^{8} + \left(98 a^{2} + 59 a + 41\right)\cdot 101^{9} +O(101^{10})\)
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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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(4,5)$ | $4$ |
| $18$ | $2$ | $(1,4)(5,8)(7,9)$ | $2$ |
| $27$ | $2$ | $(1,8)(2,3)(4,5)$ | $0$ |
| $27$ | $2$ | $(1,8)(4,5)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,8)(4,5)(6,9)$ | $2$ |
| $6$ | $3$ | $(2,3,6)$ | $0$ |
| $8$ | $3$ | $(1,8,9)(2,3,6)(4,5,7)$ | $3$ |
| $12$ | $3$ | $(2,3,6)(4,5,7)$ | $-3$ |
| $72$ | $3$ | $(1,2,4)(3,5,8)(6,7,9)$ | $0$ |
| $54$ | $4$ | $(1,4,8,5)(7,9)$ | $0$ |
| $162$ | $4$ | $(2,4,3,5)(6,7)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,3,6)(5,8)(7,9)$ | $2$ |
| $36$ | $6$ | $(2,5,3,7,6,4)$ | $-1$ |
| $36$ | $6$ | $(2,3,6)(4,5)$ | $-2$ |
| $36$ | $6$ | $(1,8,9)(2,3,6)(4,5)$ | $1$ |
| $54$ | $6$ | $(1,8)(2,6,3)(4,5)$ | $0$ |
| $72$ | $6$ | $(1,5,8,7,9,4)(2,3,6)$ | $-1$ |
| $108$ | $6$ | $(1,2,8,3,9,6)(4,5)$ | $-1$ |
| $216$ | $6$ | $(1,2,4,8,3,5)(6,7,9)$ | $0$ |
| $144$ | $9$ | $(1,2,5,8,3,7,9,6,4)$ | $0$ |
| $108$ | $12$ | $(1,4,8,5)(2,3,6)(7,9)$ | $0$ |