Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(100350148023\)\(\medspace = 3^{3} \cdot 1549^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.17271375476403.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.4647.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.17271375476403.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 5x^{6} - 6x^{5} - 33x^{4} + 35x^{3} - 77x^{2} + 39x - 7 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{3} + 2x + 68 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 38 a^{2} + 9 a + 24 + \left(20 a^{2} + 60 a + 4\right)\cdot 73 + \left(72 a^{2} + 19 a + 22\right)\cdot 73^{2} + \left(20 a^{2} + 43 a + 69\right)\cdot 73^{3} + \left(21 a^{2} + 8 a + 11\right)\cdot 73^{4} + \left(5 a^{2} + 25 a + 38\right)\cdot 73^{5} + \left(10 a^{2} + 28 a + 50\right)\cdot 73^{6} + \left(9 a^{2} + 40 a + 13\right)\cdot 73^{7} + \left(58 a^{2} + 62 a + 18\right)\cdot 73^{8} + \left(16 a^{2} + 63 a + 63\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 71 a^{2} + 22 a + 60 + \left(28 a^{2} + 33 a + 20\right)\cdot 73 + \left(37 a^{2} + 54 a + 35\right)\cdot 73^{2} + \left(11 a^{2} + 61 a + 45\right)\cdot 73^{3} + \left(47 a^{2} + 39 a + 47\right)\cdot 73^{4} + \left(32 a^{2} + 40 a + 66\right)\cdot 73^{5} + \left(5 a^{2} + 14 a + 40\right)\cdot 73^{6} + \left(5 a^{2} + 17 a + 26\right)\cdot 73^{7} + \left(60 a^{2} + 57 a + 71\right)\cdot 73^{8} + \left(10 a^{2} + 12 a + 2\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a^{2} + 30 a + 47 + \left(56 a^{2} + 45 a + 3\right)\cdot 73 + \left(31 a^{2} + 17 a + 41\right)\cdot 73^{2} + \left(2 a^{2} + 55 a + 44\right)\cdot 73^{3} + \left(46 a^{2} + 22 a + 20\right)\cdot 73^{4} + \left(48 a^{2} + 18 a + 47\right)\cdot 73^{5} + \left(66 a^{2} + 67 a + 28\right)\cdot 73^{6} + \left(10 a^{2} + 14 a + 40\right)\cdot 73^{7} + \left(55 a^{2} + 55 a + 38\right)\cdot 73^{8} + \left(31 a^{2} + 39 a + 34\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 + 69\cdot 73 + 3\cdot 73^{2} + 22\cdot 73^{3} + 49\cdot 73^{4} + 52\cdot 73^{5} + 34\cdot 73^{6} + 68\cdot 73^{7} + 31\cdot 73^{8} + 23\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 71 a^{2} + 34 a + 68 + \left(68 a^{2} + 40 a + 68\right)\cdot 73 + \left(41 a^{2} + 35 a + 5\right)\cdot 73^{2} + \left(49 a^{2} + 47 a + 10\right)\cdot 73^{3} + \left(5 a^{2} + 41 a + 64\right)\cdot 73^{4} + \left(19 a^{2} + 29 a + 7\right)\cdot 73^{5} + \left(69 a^{2} + 50 a + 32\right)\cdot 73^{6} + \left(52 a^{2} + 17 a + 23\right)\cdot 73^{7} + \left(32 a^{2} + 28 a + 57\right)\cdot 73^{8} + \left(24 a^{2} + 42 a + 24\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 + 53\cdot 73 + 43\cdot 73^{2} + 55\cdot 73^{3} + 45\cdot 73^{4} + 3\cdot 73^{5} + 45\cdot 73^{6} + 13\cdot 73^{7} + 26\cdot 73^{8} + 34\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 9 a^{2} + 4 a + 26 + \left(22 a^{2} + 4 a + 60\right)\cdot 73 + \left(65 a^{2} + 35 a + 23\right)\cdot 73^{2} + \left(14 a^{2} + 20 a + 1\right)\cdot 73^{3} + \left(67 a^{2} + 50\right)\cdot 73^{4} + \left(6 a^{2} + 64 a + 56\right)\cdot 73^{5} + \left(18 a^{2} + 33 a + 57\right)\cdot 73^{6} + \left(61 a^{2} + 59 a + 52\right)\cdot 73^{7} + \left(12 a^{2} + 65 a + 32\right)\cdot 73^{8} + \left(8 a^{2} + 11 a + 72\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 66 a^{2} + 47 a + 29 + \left(21 a^{2} + 35 a + 11\right)\cdot 73 + \left(43 a^{2} + 56 a + 43\right)\cdot 73^{2} + \left(46 a^{2} + 63 a + 43\right)\cdot 73^{3} + \left(31 a^{2} + 32 a + 2\right)\cdot 73^{4} + \left(33 a^{2} + 41 a + 19\right)\cdot 73^{5} + \left(49 a^{2} + 24 a + 2\right)\cdot 73^{6} + \left(6 a^{2} + 69 a + 53\right)\cdot 73^{7} + \left(22 a + 15\right)\cdot 73^{8} + \left(54 a^{2} + 48 a + 36\right)\cdot 73^{9} +O(73^{10})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,4)(3,5)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $1$ |
| $12$ | $2$ | $(2,6)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $3$ |
| $36$ | $2$ | $(1,3)(2,6)$ | $1$ |
| $36$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $16$ | $3$ | $(2,7,8)$ | $0$ |
| $64$ | $3$ | $(2,7,8)(3,4,5)$ | $0$ |
| $12$ | $4$ | $(1,3,4,5)$ | $-3$ |
| $36$ | $4$ | $(1,3,4,5)(2,6,7,8)$ | $1$ |
| $36$ | $4$ | $(1,4)(2,6,7,8)(3,5)$ | $1$ |
| $72$ | $4$ | $(1,7,4,2)(3,8,5,6)$ | $-1$ |
| $72$ | $4$ | $(1,3,4,5)(2,6)$ | $-1$ |
| $144$ | $4$ | $(1,2,3,6)(4,7)(5,8)$ | $1$ |
| $48$ | $6$ | $(1,4)(2,8,7)(3,5)$ | $0$ |
| $96$ | $6$ | $(2,6)(3,5,4)$ | $0$ |
| $192$ | $6$ | $(1,6)(2,3,7,4,8,5)$ | $0$ |
| $144$ | $8$ | $(1,6,3,7,4,8,5,2)$ | $-1$ |
| $96$ | $12$ | $(1,3,4,5)(2,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.