Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(220775720368322943\)\(\medspace = 3^{15} \cdot 109^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.102904299369.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | odd |
| Projective image: | $C_3^3:S_4$ |
| Projective field: | Galois closure of 9.1.102904299369.1 |
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 $\mathbb{Q}_{ 101 }$:
\( x^{3} + 3x + 99 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 27 + 11\cdot 101 + 71\cdot 101^{2} + 92\cdot 101^{3} + 97\cdot 101^{4} + 37\cdot 101^{5} + 30\cdot 101^{6} + 77\cdot 101^{7} + 70\cdot 101^{8} + 9\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 89 + 88\cdot 101 + 44\cdot 101^{2} + 93\cdot 101^{3} + 58\cdot 101^{4} + 10\cdot 101^{5} + 100\cdot 101^{6} + 57\cdot 101^{7} + 61\cdot 101^{8} + 41\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 77 + 14\cdot 101 + 24\cdot 101^{2} + 60\cdot 101^{3} + 16\cdot 101^{4} + 100\cdot 101^{5} + 42\cdot 101^{6} + 43\cdot 101^{7} + 69\cdot 101^{8} + 16\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 a^{2} + 11 a + 71 + \left(27 a^{2} + 54 a + 63\right)\cdot 101 + \left(21 a^{2} + 83 a + 62\right)\cdot 101^{2} + \left(88 a^{2} + 60 a + 29\right)\cdot 101^{3} + \left(44 a^{2} + 14 a + 76\right)\cdot 101^{4} + \left(52 a^{2} + 54 a + 35\right)\cdot 101^{5} + \left(90 a^{2} + 70 a + 85\right)\cdot 101^{6} + \left(74 a^{2} + 90 a + 86\right)\cdot 101^{7} + \left(25 a^{2} + 56 a + 39\right)\cdot 101^{8} + \left(96 a^{2} + 32 a + 15\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 69 a^{2} + a + 90 + \left(48 a^{2} + 37 a + 50\right)\cdot 101 + \left(24 a^{2} + 94 a + 49\right)\cdot 101^{2} + \left(13 a^{2} + 96 a + 57\right)\cdot 101^{3} + \left(26 a^{2} + 17 a + 41\right)\cdot 101^{4} + \left(11 a^{2} + 87 a + 8\right)\cdot 101^{5} + \left(25 a^{2} + 43 a + 88\right)\cdot 101^{6} + \left(88 a^{2} + 48 a + 78\right)\cdot 101^{7} + \left(14 a^{2} + 35 a + 7\right)\cdot 101^{8} + \left(78 a^{2} + 2 a + 75\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 52 a^{2} + 7 a + 56 + \left(87 a^{2} + 2 a + 27\right)\cdot 101 + \left(13 a^{2} + 90 a + 28\right)\cdot 101^{2} + \left(70 a^{2} + 9 a + 70\right)\cdot 101^{3} + \left(69 a^{2} + 73 a + 27\right)\cdot 101^{4} + \left(60 a^{2} + 91 a + 6\right)\cdot 101^{5} + \left(6 a^{2} + 47 a + 51\right)\cdot 101^{6} + \left(62 a^{2} + a + 26\right)\cdot 101^{7} + \left(36 a^{2} + 37 a + 51\right)\cdot 101^{8} + \left(29 a^{2} + 72 a + 78\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 a^{2} + 50 a + 85 + \left(15 a^{2} + 23 a + 39\right)\cdot 101 + \left(45 a^{2} + 17 a + 9\right)\cdot 101^{2} + \left(65 a^{2} + 91 a + 85\right)\cdot 101^{3} + \left(93 a^{2} + 76 a + 72\right)\cdot 101^{4} + \left(70 a^{2} + 99 a + 72\right)\cdot 101^{5} + \left(92 a^{2} + 37 a + 89\right)\cdot 101^{6} + \left(8 a^{2} + 43 a + 55\right)\cdot 101^{7} + \left(30 a^{2} + 46 a + 48\right)\cdot 101^{8} + \left(85 a^{2} + 79 a + 94\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 41 a^{2} + 40 a + 100 + \left(57 a^{2} + 23 a + 21\right)\cdot 101 + \left(34 a^{2} + 89\right)\cdot 101^{2} + \left(48 a^{2} + 50 a + 50\right)\cdot 101^{3} + \left(63 a^{2} + 9 a + 12\right)\cdot 101^{4} + \left(78 a^{2} + 48 a + 88\right)\cdot 101^{5} + \left(18 a^{2} + 93 a + 42\right)\cdot 101^{6} + \left(17 a^{2} + 67 a + 72\right)\cdot 101^{7} + \left(45 a^{2} + 98 a + 78\right)\cdot 101^{8} + \left(20 a^{2} + 89 a + 65\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 81 a^{2} + 93 a + 13 + \left(65 a^{2} + 61 a + 85\right)\cdot 101 + \left(62 a^{2} + 17 a + 24\right)\cdot 101^{2} + \left(17 a^{2} + 95 a + 66\right)\cdot 101^{3} + \left(5 a^{2} + 9 a + 100\right)\cdot 101^{4} + \left(29 a^{2} + 23 a + 43\right)\cdot 101^{5} + \left(69 a^{2} + 9 a + 75\right)\cdot 101^{6} + \left(51 a^{2} + 51 a + 5\right)\cdot 101^{7} + \left(49 a^{2} + 28 a + 77\right)\cdot 101^{8} + \left(94 a^{2} + 26 a + 6\right)\cdot 101^{9} +O(101^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,7)(3,8)(5,6)$ | $2$ |
| $6$ | $3$ | $(5,9,6)$ | $0$ |
| $8$ | $3$ | $(1,2,3)(4,7,8)(5,6,9)$ | $3$ |
| $12$ | $3$ | $(1,2,3)(4,7,8)$ | $-3$ |
| $72$ | $3$ | $(1,5,4)(2,6,7)(3,9,8)$ | $0$ |
| $54$ | $4$ | $(1,5,2,6)(3,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(4,7)(5,6,9)$ | $0$ |
| $108$ | $6$ | $(1,7,2,8,3,4)(5,6)$ | $-1$ |
| $72$ | $9$ | $(1,5,7,2,6,8,3,9,4)$ | $0$ |
| $72$ | $9$ | $(1,5,8,3,9,7,2,6,4)$ | $0$ |
| $54$ | $12$ | $(1,4,2,7)(3,8)(5,9,6)$ | $0$ |
| $54$ | $12$ | $(1,4,2,7)(3,8)(5,6,9)$ | $0$ |