Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(910134926371\)\(\medspace = 11^{3} \cdot 881^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.6626684877131.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.9691.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.6626684877131.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 9x^{6} - 43x^{5} + 84x^{4} - 173x^{3} + 218x^{2} - 137x + 52 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 139 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 139 }$:
\( x^{2} + 138x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 118 + 130\cdot 139 + 98\cdot 139^{2} + 27\cdot 139^{3} + 55\cdot 139^{4} + 78\cdot 139^{5} + 24\cdot 139^{6} + 5\cdot 139^{7} + 26\cdot 139^{8} + 61\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 136 a + 71 + \left(67 a + 23\right)\cdot 139 + \left(48 a + 81\right)\cdot 139^{2} + \left(7 a + 78\right)\cdot 139^{3} + \left(11 a + 114\right)\cdot 139^{4} + \left(57 a + 113\right)\cdot 139^{5} + \left(8 a + 94\right)\cdot 139^{6} + \left(106 a + 30\right)\cdot 139^{7} + \left(32 a + 45\right)\cdot 139^{8} + \left(55 a + 107\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 41 a + 119 + \left(38 a + 11\right)\cdot 139 + \left(110 a + 101\right)\cdot 139^{2} + \left(64 a + 103\right)\cdot 139^{3} + \left(73 a + 87\right)\cdot 139^{4} + \left(115 a + 50\right)\cdot 139^{5} + \left(69 a + 91\right)\cdot 139^{6} + \left(59 a + 64\right)\cdot 139^{7} + \left(133 a + 93\right)\cdot 139^{8} + \left(29 a + 2\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 85 a + 62 + \left(3 a + 52\right)\cdot 139 + \left(72 a + 125\right)\cdot 139^{2} + \left(18 a + 102\right)\cdot 139^{3} + \left(18 a + 117\right)\cdot 139^{4} + \left(a + 55\right)\cdot 139^{5} + \left(98 a + 56\right)\cdot 139^{6} + \left(70 a + 45\right)\cdot 139^{7} + \left(86 a + 18\right)\cdot 139^{8} + \left(29 a + 39\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 68 + \left(71 a + 94\right)\cdot 139 + \left(90 a + 61\right)\cdot 139^{2} + \left(131 a + 37\right)\cdot 139^{3} + \left(127 a + 118\right)\cdot 139^{4} + \left(81 a + 20\right)\cdot 139^{5} + \left(130 a + 46\right)\cdot 139^{6} + \left(32 a + 128\right)\cdot 139^{7} + \left(106 a + 110\right)\cdot 139^{8} + \left(83 a + 129\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 54 a + 8 + \left(135 a + 110\right)\cdot 139 + \left(66 a + 54\right)\cdot 139^{2} + \left(120 a + 49\right)\cdot 139^{3} + \left(120 a + 117\right)\cdot 139^{4} + \left(137 a + 38\right)\cdot 139^{5} + \left(40 a + 14\right)\cdot 139^{6} + \left(68 a + 18\right)\cdot 139^{7} + \left(52 a + 34\right)\cdot 139^{8} + \left(109 a + 121\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 98 a + 21 + \left(100 a + 9\right)\cdot 139 + \left(28 a + 34\right)\cdot 139^{2} + \left(74 a + 58\right)\cdot 139^{3} + \left(65 a + 96\right)\cdot 139^{4} + \left(23 a + 92\right)\cdot 139^{5} + \left(69 a + 45\right)\cdot 139^{6} + \left(79 a + 54\right)\cdot 139^{7} + \left(5 a + 28\right)\cdot 139^{8} + \left(109 a + 38\right)\cdot 139^{9} +O(139^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 91 + 123\cdot 139 + 137\cdot 139^{2} + 97\cdot 139^{3} + 126\cdot 139^{4} + 104\cdot 139^{5} + 43\cdot 139^{6} + 70\cdot 139^{7} + 60\cdot 139^{8} + 56\cdot 139^{9} +O(139^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,6)(4,8)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,3)(4,8)$ | $-1$ |
| $16$ | $3$ | $(2,5,7)$ | $0$ |
| $64$ | $3$ | $(2,5,7)(4,6,8)$ | $0$ |
| $12$ | $4$ | $(1,4,6,8)$ | $-3$ |
| $36$ | $4$ | $(1,4,6,8)(2,3,5,7)$ | $1$ |
| $36$ | $4$ | $(1,6)(2,3,5,7)(4,8)$ | $1$ |
| $72$ | $4$ | $(1,5,6,2)(3,4,7,8)$ | $-1$ |
| $72$ | $4$ | $(1,4,6,8)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,4,3)(5,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,6)(2,7,5)(4,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(4,8,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,4,5,6,7,8)$ | $0$ |
| $144$ | $8$ | $(1,3,4,5,6,7,8,2)$ | $-1$ |
| $96$ | $12$ | $(1,4,6,8)(2,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.