Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(111145882112\)\(\medspace = 2^{9} \cdot 601^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.1043729299208.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | even |
| Determinant: | 1.4808.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.1043729299208.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 9x^{6} + 32x^{5} + 18x^{4} - 125x^{3} + 46x^{2} + 165x - 94 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 44 + 45 a\cdot 67 + \left(18 a + 51\right)\cdot 67^{2} + \left(50 a + 32\right)\cdot 67^{3} + \left(5 a + 8\right)\cdot 67^{4} + \left(34 a + 44\right)\cdot 67^{5} + \left(17 a + 19\right)\cdot 67^{6} + \left(16 a + 24\right)\cdot 67^{7} + 32 a\cdot 67^{8} + \left(45 a + 63\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 64\cdot 67 + 46\cdot 67^{2} + 67^{3} + 13\cdot 67^{4} + 9\cdot 67^{5} + 8\cdot 67^{6} + 3\cdot 67^{7} + 53\cdot 67^{8} + 4\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 64 a + 14 + \left(10 a + 53\right)\cdot 67 + \left(3 a + 48\right)\cdot 67^{2} + \left(40 a + 28\right)\cdot 67^{3} + \left(15 a + 35\right)\cdot 67^{4} + \left(16 a + 63\right)\cdot 67^{5} + \left(10 a + 31\right)\cdot 67^{6} + \left(16 a + 11\right)\cdot 67^{7} + \left(66 a + 6\right)\cdot 67^{8} + \left(42 a + 11\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 + 41\cdot 67 + 22\cdot 67^{2} + 18\cdot 67^{3} + 64\cdot 67^{4} + 39\cdot 67^{5} + 50\cdot 67^{6} + 34\cdot 67^{7} + 34\cdot 67^{8} + 54\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 2 + \left(56 a + 33\right)\cdot 67 + \left(63 a + 50\right)\cdot 67^{2} + \left(26 a + 51\right)\cdot 67^{3} + \left(51 a + 57\right)\cdot 67^{4} + \left(50 a + 45\right)\cdot 67^{5} + \left(56 a + 56\right)\cdot 67^{6} + \left(50 a + 65\right)\cdot 67^{7} + 53\cdot 67^{8} + \left(24 a + 49\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 20 + \left(13 a + 66\right)\cdot 67 + \left(21 a + 14\right)\cdot 67^{2} + \left(6 a + 58\right)\cdot 67^{3} + \left(46 a + 31\right)\cdot 67^{4} + \left(23 a + 21\right)\cdot 67^{5} + \left(46 a + 42\right)\cdot 67^{6} + 16\cdot 67^{7} + \left(19 a + 66\right)\cdot 67^{8} + \left(29 a + 20\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 47 a + 57 + \left(21 a + 27\right)\cdot 67 + \left(48 a + 13\right)\cdot 67^{2} + \left(16 a + 14\right)\cdot 67^{3} + \left(61 a + 48\right)\cdot 67^{4} + \left(32 a + 40\right)\cdot 67^{5} + \left(49 a + 55\right)\cdot 67^{6} + \left(50 a + 4\right)\cdot 67^{7} + \left(34 a + 46\right)\cdot 67^{8} + \left(21 a + 11\right)\cdot 67^{9} +O(67^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 64 a + 32 + \left(53 a + 48\right)\cdot 67 + \left(45 a + 19\right)\cdot 67^{2} + \left(60 a + 62\right)\cdot 67^{3} + \left(20 a + 8\right)\cdot 67^{4} + \left(43 a + 3\right)\cdot 67^{5} + \left(20 a + 3\right)\cdot 67^{6} + \left(66 a + 40\right)\cdot 67^{7} + \left(47 a + 7\right)\cdot 67^{8} + \left(37 a + 52\right)\cdot 67^{9} +O(67^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,4)(2,7)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $1$ |
| $12$ | $2$ | $(3,5)$ | $3$ |
| $24$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,5)$ | $1$ |
| $36$ | $2$ | $(1,4)(2,7)(3,5)$ | $-1$ |
| $16$ | $3$ | $(3,6,8)$ | $0$ |
| $64$ | $3$ | $(2,4,7)(3,6,8)$ | $0$ |
| $12$ | $4$ | $(1,2,4,7)$ | $-3$ |
| $36$ | $4$ | $(1,2,4,7)(3,5,6,8)$ | $1$ |
| $36$ | $4$ | $(1,4)(2,7)(3,5,6,8)$ | $1$ |
| $72$ | $4$ | $(1,6,4,3)(2,8,7,5)$ | $-1$ |
| $72$ | $4$ | $(1,2,4,7)(3,5)$ | $-1$ |
| $144$ | $4$ | $(1,3,2,5)(4,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,4)(2,7)(3,8,6)$ | $0$ |
| $96$ | $6$ | $(2,7,4)(3,5)$ | $0$ |
| $192$ | $6$ | $(1,5)(2,6,4,8,7,3)$ | $0$ |
| $144$ | $8$ | $(1,5,2,6,4,8,7,3)$ | $-1$ |
| $96$ | $12$ | $(1,2,4,7)(3,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.