Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr S_4$ |
| Conductor: | \(15432\)\(\medspace = 2^{3} \cdot 3 \cdot 643 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.29768328.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr S_4$ |
| Parity: | even |
| Determinant: | 1.15432.2t1.b.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.4.15241383936.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{2} + 82x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 81\cdot 83 + 54\cdot 83^{2} + 46\cdot 83^{3} + 12\cdot 83^{4} + 2\cdot 83^{5} + 39\cdot 83^{6} + 58\cdot 83^{7} + 30\cdot 83^{8} + 45\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 50 a + 18 + \left(34 a + 73\right)\cdot 83 + \left(31 a + 58\right)\cdot 83^{2} + \left(79 a + 18\right)\cdot 83^{3} + \left(35 a + 39\right)\cdot 83^{4} + \left(66 a + 51\right)\cdot 83^{5} + \left(26 a + 45\right)\cdot 83^{6} + \left(14 a + 9\right)\cdot 83^{7} + \left(41 a + 60\right)\cdot 83^{8} + \left(26 a + 54\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 68 + \left(48 a + 57\right)\cdot 83 + \left(51 a + 55\right)\cdot 83^{2} + \left(3 a + 66\right)\cdot 83^{3} + \left(47 a + 78\right)\cdot 83^{4} + \left(16 a + 81\right)\cdot 83^{5} + \left(56 a + 5\right)\cdot 83^{6} + \left(68 a + 80\right)\cdot 83^{7} + \left(41 a + 3\right)\cdot 83^{8} + \left(56 a + 40\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 77 + 16\cdot 83 + 8\cdot 83^{2} + 61\cdot 83^{3} + 32\cdot 83^{4} + 26\cdot 83^{5} + 43\cdot 83^{6} + 15\cdot 83^{7} + 61\cdot 83^{8} + 72\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a + 45 + \left(41 a + 28\right)\cdot 83 + \left(35 a + 37\right)\cdot 83^{2} + \left(41 a + 7\right)\cdot 83^{3} + \left(75 a + 61\right)\cdot 83^{4} + \left(75 a + 29\right)\cdot 83^{5} + \left(41 a + 79\right)\cdot 83^{6} + \left(13 a + 11\right)\cdot 83^{7} + \left(81 a + 10\right)\cdot 83^{8} + \left(19 a + 17\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 a + 64 + \left(41 a + 50\right)\cdot 83 + \left(47 a + 31\right)\cdot 83^{2} + \left(41 a + 13\right)\cdot 83^{3} + \left(7 a + 12\right)\cdot 83^{4} + \left(7 a + 30\right)\cdot 83^{5} + \left(41 a + 45\right)\cdot 83^{6} + \left(69 a + 66\right)\cdot 83^{7} + \left(a + 77\right)\cdot 83^{8} + \left(63 a + 38\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a + 59 + \left(26 a + 4\right)\cdot 83 + \left(58 a + 68\right)\cdot 83^{2} + \left(20 a + 77\right)\cdot 83^{3} + \left(76 a + 19\right)\cdot 83^{4} + \left(69 a + 58\right)\cdot 83^{5} + \left(71 a + 35\right)\cdot 83^{6} + \left(3 a + 37\right)\cdot 83^{7} + \left(17 a + 37\right)\cdot 83^{8} + \left(30 a + 66\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 71 a + 71 + \left(56 a + 18\right)\cdot 83 + \left(24 a + 17\right)\cdot 83^{2} + \left(62 a + 40\right)\cdot 83^{3} + \left(6 a + 75\right)\cdot 83^{4} + \left(13 a + 51\right)\cdot 83^{5} + \left(11 a + 37\right)\cdot 83^{6} + \left(79 a + 52\right)\cdot 83^{7} + \left(65 a + 50\right)\cdot 83^{8} + \left(52 a + 79\right)\cdot 83^{9} +O(83^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $-4$ |
| $4$ | $2$ | $(1,4)$ | $2$ |
| $4$ | $2$ | $(1,4)(2,3)(6,7)$ | $-2$ |
| $6$ | $2$ | $(1,4)(5,8)$ | $0$ |
| $12$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $12$ | $2$ | $(2,5)(3,8)$ | $2$ |
| $12$ | $2$ | $(1,6)(2,3)(4,7)(5,8)$ | $-2$ |
| $24$ | $2$ | $(1,4)(2,5)(3,8)$ | $0$ |
| $32$ | $3$ | $(1,2,6)(3,7,4)$ | $1$ |
| $12$ | $4$ | $(1,8,4,5)(2,6,3,7)$ | $0$ |
| $12$ | $4$ | $(2,5,3,8)$ | $2$ |
| $12$ | $4$ | $(1,7,4,6)(2,3)(5,8)$ | $-2$ |
| $24$ | $4$ | $(1,8,4,5)(2,6)(3,7)$ | $0$ |
| $24$ | $4$ | $(1,4)(2,5,3,8)$ | $0$ |
| $48$ | $4$ | $(1,2,5,6)(3,8,7,4)$ | $0$ |
| $32$ | $6$ | $(1,3,7,4,2,6)$ | $1$ |
| $32$ | $6$ | $(1,2,6)(3,7,4)(5,8)$ | $-1$ |
| $32$ | $6$ | $(1,3,7,4,2,6)(5,8)$ | $-1$ |
| $48$ | $8$ | $(1,3,8,7,4,2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.