Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(12786520101630016\)\(\medspace = 2^{6} \cdot 7^{10} \cdot 29^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.5.10699470656.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Projective image: | $S_3\wr C_3$ |
| Projective field: | Galois closure of 9.5.10699470656.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{3} + 3x + 81 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a^{2} + 67 a + 61 + \left(32 a^{2} + 82 a + 34\right)\cdot 83 + \left(35 a^{2} + 11 a + 33\right)\cdot 83^{2} + \left(66 a^{2} + 63 a + 7\right)\cdot 83^{3} + \left(52 a^{2} + 73 a + 47\right)\cdot 83^{4} + \left(11 a^{2} + 10 a + 3\right)\cdot 83^{5} + \left(33 a^{2} + 42 a + 7\right)\cdot 83^{6} + \left(80 a^{2} + 2 a + 60\right)\cdot 83^{7} + \left(20 a^{2} + 41 a + 81\right)\cdot 83^{8} + \left(32 a^{2} + 29 a + 57\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a^{2} + 77 a + 75 + \left(82 a^{2} + 62 a + 32\right)\cdot 83 + \left(2 a^{2} + 25 a + 7\right)\cdot 83^{2} + \left(78 a^{2} + 27 a + 10\right)\cdot 83^{3} + \left(66 a^{2} + 48 a + 49\right)\cdot 83^{4} + \left(61 a^{2} + 33 a + 66\right)\cdot 83^{5} + \left(60 a^{2} + 26 a + 6\right)\cdot 83^{6} + \left(2 a^{2} + 3 a + 58\right)\cdot 83^{7} + \left(62 a^{2} + 34 a + 56\right)\cdot 83^{8} + \left(37 a^{2} + 77 a + 74\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 61 a^{2} + 11 a + 15 + \left(39 a^{2} + 60 a + 50\right)\cdot 83 + \left(50 a^{2} + 45 a + 63\right)\cdot 83^{2} + \left(43 a + 41\right)\cdot 83^{3} + \left(66 a^{2} + 4 a + 73\right)\cdot 83^{4} + \left(21 a^{2} + 82 a + 23\right)\cdot 83^{5} + \left(62 a^{2} + 63 a + 65\right)\cdot 83^{6} + \left(82 a^{2} + 73 a + 64\right)\cdot 83^{7} + \left(31 a^{2} + 50 a + 20\right)\cdot 83^{8} + \left(60 a^{2} + 39 a + 31\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 a^{2} + 5 a + 18 + \left(11 a^{2} + 23 a + 76\right)\cdot 83 + \left(80 a^{2} + 25 a + 39\right)\cdot 83^{2} + \left(15 a^{2} + 59 a + 72\right)\cdot 83^{3} + \left(47 a^{2} + 4 a + 35\right)\cdot 83^{4} + \left(49 a^{2} + 73 a + 79\right)\cdot 83^{5} + \left(70 a^{2} + 59 a + 81\right)\cdot 83^{6} + \left(2 a^{2} + 6 a + 70\right)\cdot 83^{7} + \left(30 a^{2} + 74 a + 16\right)\cdot 83^{8} + \left(73 a^{2} + 13 a + 57\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 51 + 49\cdot 83 + 32\cdot 83^{2} + 29\cdot 83^{3} + 48\cdot 83^{4} + 17\cdot 83^{5} + 72\cdot 83^{6} + 39\cdot 83^{7} + 21\cdot 83^{8} + 10\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 72 a^{2} + 71 a + 25 + \left(42 a^{2} + 64 a + 37\right)\cdot 83 + \left(71 a + 2\right)\cdot 83^{2} + \left(19 a^{2} + 39 a + 58\right)\cdot 83^{3} + \left(75 a^{2} + 57 a + 65\right)\cdot 83^{4} + \left(2 a^{2} + 82 a + 31\right)\cdot 83^{5} + \left(23 a^{2} + 9 a + 14\right)\cdot 83^{6} + \left(63 a^{2} + 76 a + 13\right)\cdot 83^{7} + \left(33 a^{2} + 9 a\right)\cdot 83^{8} + \left(19 a^{2} + 10 a + 38\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 36 + 53\cdot 83 + 19\cdot 83^{2} + 80\cdot 83^{3} + 10\cdot 83^{4} + 59\cdot 83^{5} + 34\cdot 83^{6} + 38\cdot 83^{7} + 16\cdot 83^{8} + 48\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 + 47\cdot 83 + 55\cdot 83^{2} + 40\cdot 83^{3} + 38\cdot 83^{4} + 70\cdot 83^{5} + 82\cdot 83^{6} + 65\cdot 83^{7} + 44\cdot 83^{8} + 46\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 80 a^{2} + 18 a + 41 + \left(40 a^{2} + 38 a + 33\right)\cdot 83 + \left(79 a^{2} + 68 a + 77\right)\cdot 83^{2} + \left(68 a^{2} + 15 a + 74\right)\cdot 83^{3} + \left(23 a^{2} + 60 a + 45\right)\cdot 83^{4} + \left(18 a^{2} + 49 a + 62\right)\cdot 83^{5} + \left(82 a^{2} + 46 a + 49\right)\cdot 83^{6} + \left(16 a^{2} + 3 a + 3\right)\cdot 83^{7} + \left(70 a^{2} + 39 a + 73\right)\cdot 83^{8} + \left(25 a^{2} + 78 a + 50\right)\cdot 83^{9} +O(83^{10})\)
|
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(5,7)$ | $4$ |
| $27$ | $2$ | $(1,3)(2,6)(5,7)$ | $0$ |
| $27$ | $2$ | $(3,4)(5,7)$ | $0$ |
| $6$ | $3$ | $(2,6,9)$ | $0$ |
| $8$ | $3$ | $(1,3,4)(2,6,9)(5,7,8)$ | $3$ |
| $12$ | $3$ | $(1,3,4)(2,6,9)$ | $-3$ |
| $36$ | $3$ | $(1,5,2)(3,7,6)(4,8,9)$ | $0$ |
| $36$ | $3$ | $(1,2,5)(3,6,7)(4,9,8)$ | $0$ |
| $18$ | $6$ | $(2,6,9)(5,7)$ | $-2$ |
| $18$ | $6$ | $(1,3,4)(5,7)$ | $-2$ |
| $36$ | $6$ | $(1,3,4)(2,6,9)(5,7)$ | $1$ |
| $54$ | $6$ | $(2,6,9)(3,4)(5,7)$ | $0$ |
| $108$ | $6$ | $(1,5,6,3,7,2)(4,8,9)$ | $0$ |
| $108$ | $6$ | $(1,2,7,3,6,5)(4,9,8)$ | $0$ |
| $72$ | $9$ | $(1,5,2,3,7,6,4,8,9)$ | $0$ |
| $72$ | $9$ | $(1,2,7,4,9,5,3,6,8)$ | $0$ |