Basic invariants
| Dimension: | $6$ |
| Group: | $V_4^2:(S_3\times C_2)$ |
| Conductor: | \(292887232721\)\(\medspace = 29^{3} \cdot 229^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.37090522921.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $V_4^2:(S_3\times C_2)$ |
| Parity: | even |
| Determinant: | 1.6641.2t1.a.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.4.37090522921.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + x^{6} - 4x^{5} + 11x^{4} + 8x^{3} - 16x^{2} + 2x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$:
\( x^{3} + 4x + 64 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 53 a^{2} + 23 a + 44 + \left(59 a^{2} + 11 a + 20\right)\cdot 71 + \left(6 a^{2} + 54 a + 33\right)\cdot 71^{2} + \left(36 a^{2} + 2 a + 14\right)\cdot 71^{3} + \left(32 a^{2} + 66 a + 27\right)\cdot 71^{4} + \left(18 a^{2} + 70 a + 42\right)\cdot 71^{5} + \left(15 a^{2} + 39 a + 25\right)\cdot 71^{6} + \left(34 a^{2} + a + 27\right)\cdot 71^{7} + \left(54 a^{2} + 58 a + 33\right)\cdot 71^{8} + \left(33 a^{2} + 45 a + 49\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 a^{2} + 41 a + 67 + \left(30 a^{2} + 56 a + 60\right)\cdot 71 + \left(62 a^{2} + 58 a + 15\right)\cdot 71^{2} + \left(52 a^{2} + 36 a + 59\right)\cdot 71^{3} + \left(7 a^{2} + 41 a + 55\right)\cdot 71^{4} + \left(a^{2} + 17 a + 19\right)\cdot 71^{5} + \left(48 a^{2} + 62 a + 18\right)\cdot 71^{6} + \left(13 a^{2} + a + 67\right)\cdot 71^{7} + \left(52 a^{2} + 3 a + 50\right)\cdot 71^{8} + \left(3 a^{2} + 8 a + 16\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 40 + 41\cdot 71 + 42\cdot 71^{2} + 37\cdot 71^{3} + 16\cdot 71^{4} + 16\cdot 71^{5} + 32\cdot 71^{6} + 23\cdot 71^{7} + 17\cdot 71^{8} + 28\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a^{2} + 58 a + 14 + \left(15 a^{2} + 51 a + 2\right)\cdot 71 + \left(40 a^{2} + 35 a + 61\right)\cdot 71^{2} + \left(59 a^{2} + 69 a + 68\right)\cdot 71^{3} + \left(42 a^{2} + 47 a + 10\right)\cdot 71^{4} + \left(43 a^{2} + 68 a + 59\right)\cdot 71^{5} + \left(2 a^{2} + 37 a + 62\right)\cdot 71^{6} + \left(53 a^{2} + 68 a + 43\right)\cdot 71^{7} + \left(53 a^{2} + 7 a + 59\right)\cdot 71^{8} + \left(23 a^{2} + 21 a + 14\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 52 a^{2} + 20 a + 47 + \left(41 a^{2} + 64 a + 25\right)\cdot 71 + \left(63 a^{2} + 23 a + 52\right)\cdot 71^{2} + \left(13 a^{2} + 44 a + 41\right)\cdot 71^{3} + \left(68 a^{2} + 58 a + 7\right)\cdot 71^{4} + \left(32 a^{2} + 18 a + 7\right)\cdot 71^{5} + \left(45 a^{2} + 63 a + 35\right)\cdot 71^{6} + \left(7 a^{2} + 10 a + 17\right)\cdot 71^{7} + \left(47 a^{2} + 14 a + 18\right)\cdot 71^{8} + \left(29 a^{2} + 39 a + 54\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 63\cdot 71 + 50\cdot 71^{2} + 51\cdot 71^{3} + 45\cdot 71^{4} + 33\cdot 71^{5} + 58\cdot 71^{6} + 34\cdot 71^{7} + 71^{8} + 26\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 54 a^{2} + 7 a + 23 + \left(51 a^{2} + 3 a + 70\right)\cdot 71 + \left(a^{2} + 29 a + 66\right)\cdot 71^{2} + \left(53 a^{2} + 31 a + 35\right)\cdot 71^{3} + \left(30 a^{2} + 34 a + 22\right)\cdot 71^{4} + \left(51 a^{2} + 53 a + 59\right)\cdot 71^{5} + \left(7 a^{2} + 39 a + 52\right)\cdot 71^{6} + \left(23 a^{2} + 67 a + 68\right)\cdot 71^{7} + \left(35 a^{2} + 9 a + 5\right)\cdot 71^{8} + \left(33 a^{2} + 17 a + 25\right)\cdot 71^{9} +O(71^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 6 a^{2} + 64 a + 19 + \left(14 a^{2} + 25 a + 70\right)\cdot 71 + \left(38 a^{2} + 11 a + 31\right)\cdot 71^{2} + \left(68 a^{2} + 28 a + 45\right)\cdot 71^{3} + \left(30 a^{2} + 35 a + 26\right)\cdot 71^{4} + \left(65 a^{2} + 54 a + 46\right)\cdot 71^{5} + \left(22 a^{2} + 40 a + 69\right)\cdot 71^{6} + \left(10 a^{2} + 62 a\right)\cdot 71^{7} + \left(41 a^{2} + 48 a + 26\right)\cdot 71^{8} + \left(17 a^{2} + 10 a + 69\right)\cdot 71^{9} +O(71^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $3$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $6$ | $2$ | $(1,7)(2,3)(4,8)(5,6)$ | $2$ |
| $6$ | $2$ | $(4,5)(6,8)$ | $-2$ |
| $12$ | $2$ | $(2,3)(4,6)$ | $0$ |
| $12$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $2$ |
| $32$ | $3$ | $(1,7,3)(5,8,6)$ | $0$ |
| $12$ | $4$ | $(1,3,2,7)(4,6,5,8)$ | $0$ |
| $12$ | $4$ | $(1,4,2,5)(3,8,7,6)$ | $0$ |
| $12$ | $4$ | $(1,4,7,6)(2,8,3,5)$ | $-2$ |
| $24$ | $4$ | $(3,7)(4,6,5,8)$ | $0$ |
| $24$ | $4$ | $(1,5,2,8)(3,6,7,4)$ | $0$ |
| $32$ | $6$ | $(1,6,7,5,3,8)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.