Basic invariants
| Dimension: | $6$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(35978634432\)\(\medspace = 2^{6} \cdot 3^{9} \cdot 13^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.7.6080389219008.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3 \wr C_3 $ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_3\wr C_3$ |
| Projective stem field: | Galois closure of 9.7.6080389219008.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 9x^{7} - 12x^{6} + 27x^{5} + 72x^{4} - 31x^{3} - 108x^{2} + 12x + 40 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{3} + 9x + 76 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 52 a^{2} + 45 a + 75 + \left(14 a^{2} + 75 a + 8\right)\cdot 79 + \left(13 a^{2} + 14 a\right)\cdot 79^{2} + \left(27 a^{2} + 6 a + 5\right)\cdot 79^{3} + \left(a^{2} + 46 a + 8\right)\cdot 79^{4} + \left(36 a^{2} + a + 58\right)\cdot 79^{5} + \left(7 a^{2} + 47 a + 44\right)\cdot 79^{6} + \left(64 a^{2} + 3 a + 68\right)\cdot 79^{7} + \left(30 a^{2} + 13 a + 26\right)\cdot 79^{8} + \left(41 a^{2} + 11 a + 11\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 74 + 60\cdot 79 + 50\cdot 79^{2} + 10\cdot 79^{3} + 74\cdot 79^{4} + 5\cdot 79^{5} + 12\cdot 79^{6} + 7\cdot 79^{7} + 46\cdot 79^{8} + 30\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a^{2} + 3 a + 65 + \left(76 a^{2} + 67 a + 62\right)\cdot 79 + \left(3 a^{2} + 17 a + 23\right)\cdot 79^{2} + \left(15 a^{2} + 6 a + 11\right)\cdot 79^{3} + \left(7 a^{2} + 54 a + 43\right)\cdot 79^{4} + \left(16 a^{2} + 59 a + 17\right)\cdot 79^{5} + \left(34 a^{2} + 50 a + 47\right)\cdot 79^{6} + \left(23 a^{2} + 36 a + 61\right)\cdot 79^{7} + \left(64 a^{2} + 25 a + 69\right)\cdot 79^{8} + \left(9 a^{2} + 39 a + 58\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a^{2} + 56 a + 5 + \left(58 a^{2} + 41 a + 33\right)\cdot 79 + \left(39 a^{2} + 70 a + 1\right)\cdot 79^{2} + \left(18 a^{2} + 69 a + 32\right)\cdot 79^{3} + \left(6 a^{2} + 8 a + 37\right)\cdot 79^{4} + \left(71 a^{2} + 42 a + 31\right)\cdot 79^{5} + \left(30 a^{2} + 27\right)\cdot 79^{6} + \left(44 a^{2} + 28 a + 29\right)\cdot 79^{7} + \left(58 a^{2} + 74 a + 35\right)\cdot 79^{8} + \left(66 a^{2} + 45 a + 5\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 73 a^{2} + 68 a + 43 + \left(50 a^{2} + 30 a + 68\right)\cdot 79 + \left(59 a^{2} + 41 a + 41\right)\cdot 79^{2} + \left(37 a^{2} + 34 a + 68\right)\cdot 79^{3} + \left(60 a^{2} + 50 a + 46\right)\cdot 79^{4} + \left(71 a^{2} + 52 a + 35\right)\cdot 79^{5} + \left(28 a^{2} + 48 a + 15\right)\cdot 79^{6} + \left(43 a^{2} + 31 a + 23\right)\cdot 79^{7} + \left(35 a^{2} + 59 a + 55\right)\cdot 79^{8} + \left(2 a^{2} + 15 a + 14\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 13 a^{2} + 57 a + 78 + \left(6 a^{2} + 40 a + 36\right)\cdot 79 + \left(26 a^{2} + 72 a + 77\right)\cdot 79^{2} + \left(33 a^{2} + 2 a + 41\right)\cdot 79^{3} + \left(71 a^{2} + 24 a + 33\right)\cdot 79^{4} + \left(50 a^{2} + 35 a + 68\right)\cdot 79^{5} + \left(40 a^{2} + 31 a + 6\right)\cdot 79^{6} + \left(49 a^{2} + 47 a + 60\right)\cdot 79^{7} + \left(68 a^{2} + 70 a + 16\right)\cdot 79^{8} + \left(49 a^{2} + 21 a + 62\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 61 a^{2} + 8 a + 50 + \left(30 a^{2} + 60 a + 26\right)\cdot 79 + \left(15 a^{2} + 19 a + 13\right)\cdot 79^{2} + \left(26 a^{2} + 38 a + 78\right)\cdot 79^{3} + \left(11 a^{2} + 53 a + 67\right)\cdot 79^{4} + \left(70 a^{2} + 45 a + 25\right)\cdot 79^{5} + \left(15 a^{2} + 58 a + 16\right)\cdot 79^{6} + \left(12 a^{2} + 10 a + 73\right)\cdot 79^{7} + \left(58 a^{2} + 73 a + 32\right)\cdot 79^{8} + \left(66 a^{2} + 23 a + 5\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 40 + 67\cdot 79 + 74\cdot 79^{2} + 63\cdot 79^{3} + 28\cdot 79^{5} + 60\cdot 79^{6} + 61\cdot 79^{7} + 68\cdot 79^{8} + 2\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 44 + 29\cdot 79 + 32\cdot 79^{2} + 4\cdot 79^{3} + 4\cdot 79^{4} + 45\cdot 79^{5} + 6\cdot 79^{6} + 10\cdot 79^{7} + 43\cdot 79^{8} + 45\cdot 79^{9} +O(79^{10})\)
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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 value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,8)$ | $4$ |
| $27$ | $2$ | $(1,4)(2,8)(3,5)$ | $0$ |
| $27$ | $2$ | $(2,8)(4,6)$ | $2$ |
| $6$ | $3$ | $(3,5,7)$ | $3$ |
| $8$ | $3$ | $(1,4,6)(2,8,9)(3,5,7)$ | $-3$ |
| $12$ | $3$ | $(1,4,6)(3,5,7)$ | $0$ |
| $36$ | $3$ | $(1,2,3)(4,8,5)(6,9,7)$ | $0$ |
| $36$ | $3$ | $(1,3,2)(4,5,8)(6,7,9)$ | $0$ |
| $18$ | $6$ | $(2,8)(3,5,7)$ | $1$ |
| $18$ | $6$ | $(1,4,6)(2,8)$ | $1$ |
| $36$ | $6$ | $(1,4,6)(2,8)(3,5,7)$ | $-2$ |
| $54$ | $6$ | $(2,8)(3,5,7)(4,6)$ | $-1$ |
| $108$ | $6$ | $(1,2,5,4,8,3)(6,9,7)$ | $0$ |
| $108$ | $6$ | $(1,3,8,4,5,2)(6,7,9)$ | $0$ |
| $72$ | $9$ | $(1,2,3,4,8,5,6,9,7)$ | $0$ |
| $72$ | $9$ | $(1,3,8,6,7,2,4,5,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.