Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(38149546589908416\)\(\medspace = 2^{6} \cdot 3^{6} \cdot 13^{4} \cdot 31^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.6915325248.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.6915325248.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 80 a^{2} + 68 a + 10 + \left(87 a^{2} + 40 a + 52\right)\cdot 101 + \left(36 a^{2} + 16 a + 29\right)\cdot 101^{2} + \left(26 a^{2} + 92 a + 48\right)\cdot 101^{3} + \left(59 a^{2} + 70 a + 77\right)\cdot 101^{4} + \left(5 a^{2} + 16 a + 36\right)\cdot 101^{5} + \left(17 a^{2} + 66 a + 25\right)\cdot 101^{6} + \left(18 a^{2} + 40\right)\cdot 101^{7} + \left(100 a^{2} + 20 a + 21\right)\cdot 101^{8} + \left(21 a^{2} + 79 a + 51\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 50 a^{2} + 28 a + 86 + \left(63 a^{2} + 80 a + 36\right)\cdot 101 + \left(93 a^{2} + a + 93\right)\cdot 101^{2} + \left(52 a^{2} + 94 a + 12\right)\cdot 101^{3} + \left(93 a^{2} + 45 a + 88\right)\cdot 101^{4} + \left(57 a^{2} + 55 a + 18\right)\cdot 101^{5} + \left(38 a^{2} + 33\right)\cdot 101^{6} + \left(61 a^{2} + 69 a + 38\right)\cdot 101^{7} + \left(16 a^{2} + 84 a + 30\right)\cdot 101^{8} + \left(56 a^{2} + 32 a + 21\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 19 a^{2} + 80 a + 90 + \left(65 a^{2} + 54 a + 6\right)\cdot 101 + \left(33 a^{2} + 28 a + 23\right)\cdot 101^{2} + \left(87 a^{2} + 73 a + 69\right)\cdot 101^{3} + \left(72 a^{2} + 38 a + 3\right)\cdot 101^{4} + \left(55 a^{2} + 30 a + 36\right)\cdot 101^{5} + \left(8 a^{2} + 97 a + 8\right)\cdot 101^{6} + \left(22 a^{2} + 59 a + 48\right)\cdot 101^{7} + \left(79 a^{2} + 29 a + 80\right)\cdot 101^{8} + \left(73 a^{2} + 10 a + 53\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 100 a^{2} + 57 a + 85 + \left(73 a^{2} + 98 a + 57\right)\cdot 101 + \left(92 a^{2} + 5 a + 91\right)\cdot 101^{2} + \left(17 a^{2} + 74 a + 43\right)\cdot 101^{3} + \left(8 a^{2} + 71 a + 18\right)\cdot 101^{4} + \left(66 a^{2} + 50 a + 35\right)\cdot 101^{5} + \left(89 a^{2} + 16 a + 34\right)\cdot 101^{6} + \left(76 a^{2} + 84 a + 69\right)\cdot 101^{7} + \left(8 a^{2} + 15 a + 14\right)\cdot 101^{8} + \left(34 a^{2} + 39 a + 78\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 52 a^{2} + 16 a + 90 + \left(64 a^{2} + 23 a + 38\right)\cdot 101 + \left(15 a^{2} + 93 a + 38\right)\cdot 101^{2} + \left(30 a^{2} + 33 a + 68\right)\cdot 101^{3} + \left(100 a^{2} + 84 a\right)\cdot 101^{4} + \left(77 a^{2} + 95 a + 59\right)\cdot 101^{5} + \left(73 a^{2} + 83 a + 2\right)\cdot 101^{6} + \left(63 a^{2} + 48 a + 43\right)\cdot 101^{7} + \left(75 a^{2} + 47\right)\cdot 101^{8} + \left(10 a^{2} + 29 a + 31\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 2 a^{2} + 54 a + 56 + \left(49 a^{2} + 5 a + 75\right)\cdot 101 + \left(30 a^{2} + 56 a + 16\right)\cdot 101^{2} + \left(88 a^{2} + 36 a + 71\right)\cdot 101^{3} + \left(69 a^{2} + 92 a + 98\right)\cdot 101^{4} + \left(39 a^{2} + 53 a + 3\right)\cdot 101^{5} + \left(75 a^{2} + 38 a + 41\right)\cdot 101^{6} + \left(60 a^{2} + 40 a + 24\right)\cdot 101^{7} + \left(22 a^{2} + 51 a + 68\right)\cdot 101^{8} + \left(5 a^{2} + 11 a + 17\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 54 a^{2} + 52 a + 3 + \left(77 a^{2} + 5 a + 32\right)\cdot 101 + \left(68 a^{2} + 81 a + 6\right)\cdot 101^{2} + \left(19 a^{2} + 94 a + 2\right)\cdot 101^{3} + \left(79 a^{2} + 52 a + 29\right)\cdot 101^{4} + \left(85 a^{2} + 26 a + 7\right)\cdot 101^{5} + \left(80 a^{2} + 89 a + 46\right)\cdot 101^{6} + \left(43 a^{2} + 59 a + 33\right)\cdot 101^{7} + \left(9 a^{2} + 63 a + 32\right)\cdot 101^{8} + \left(48 a^{2} + 10 a + 45\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 81 a^{2} + 7 a + 57 + \left(62 a^{2} + 89 a + 2\right)\cdot 101 + \left(100 a^{2} + 22 a + 70\right)\cdot 101^{2} + \left(75 a^{2} + 48 a + 13\right)\cdot 101^{3} + \left(93 a^{2} + 24 a + 58\right)\cdot 101^{4} + \left(26 a^{2} + 48 a + 91\right)\cdot 101^{5} + \left(8 a^{2} + 54 a + 1\right)\cdot 101^{6} + \left(54 a^{2} + 90 a + 54\right)\cdot 101^{7} + \left(62 a^{2} + 16 a + 37\right)\cdot 101^{8} + \left(59 a^{2} + 83 a + 68\right)\cdot 101^{9} +O(101^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 67 a^{2} + 42 a + 29 + \left(61 a^{2} + 6 a\right)\cdot 101 + \left(32 a^{2} + 98 a + 35\right)\cdot 101^{2} + \left(5 a^{2} + 58 a + 74\right)\cdot 101^{3} + \left(29 a^{2} + 23 a + 29\right)\cdot 101^{4} + \left(89 a^{2} + 26 a + 14\right)\cdot 101^{5} + \left(11 a^{2} + 58 a + 9\right)\cdot 101^{6} + \left(3 a^{2} + 51 a + 53\right)\cdot 101^{7} + \left(29 a^{2} + 20 a + 71\right)\cdot 101^{8} + \left(94 a^{2} + 7 a + 36\right)\cdot 101^{9} +O(101^{10})\)
| |
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(1,2)$ | $4$ |
| $18$ | $2$ | $(1,3)(2,4)(7,8)$ | $2$ |
| $27$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,6)(3,4)(8,9)$ | $2$ |
| $6$ | $3$ | $(5,6,9)$ | $0$ |
| $8$ | $3$ | $(1,8,2)(3,7,4)(5,9,6)$ | $3$ |
| $12$ | $3$ | $(1,8,2)(5,9,6)$ | $-3$ |
| $72$ | $3$ | $(1,3,5)(2,4,6)(7,9,8)$ | $0$ |
| $54$ | $4$ | $(1,5,2,6)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,5,2,6)(3,4)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,4)(5,6,9)(7,8)$ | $2$ |
| $36$ | $6$ | $(1,5,8,9,2,6)$ | $-1$ |
| $36$ | $6$ | $(1,2)(5,6,9)$ | $-2$ |
| $36$ | $6$ | $(1,2)(3,4,7)(5,6,9)$ | $1$ |
| $54$ | $6$ | $(1,2)(3,4)(5,9,6)$ | $0$ |
| $72$ | $6$ | $(1,3,8,7,2,4)(5,6,9)$ | $-1$ |
| $108$ | $6$ | $(1,5,8,9,2,6)(3,4)$ | $-1$ |
| $216$ | $6$ | $(1,3,5,2,4,6)(7,9,8)$ | $0$ |
| $144$ | $9$ | $(1,3,5,8,7,9,2,4,6)$ | $0$ |
| $108$ | $12$ | $(1,3,2,4)(5,6,9)(7,8)$ | $0$ |