Basic invariants
| Dimension: | $6$ |
| Group: | $V_4^2:(S_3\times C_2)$ |
| Conductor: | \(48827236375\)\(\medspace = 5^{3} \cdot 17^{3} \cdot 43^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.333975625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T108 |
| Parity: | odd |
| Determinant: | 1.3655.2t1.a.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.4.333975625.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 3x^{6} + 8x^{5} - 3x^{4} - 8x^{3} + 14x^{2} - 8x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{3} + 5x + 57 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 a^{2} + 50 a + 39 + \left(57 a^{2} + 52 a + 45\right)\cdot 59 + \left(56 a^{2} + 38 a + 47\right)\cdot 59^{2} + \left(12 a^{2} + 38 a + 33\right)\cdot 59^{3} + \left(55 a^{2} + 5 a + 49\right)\cdot 59^{4} + \left(32 a^{2} + a + 39\right)\cdot 59^{5} + \left(3 a^{2} + 39 a + 42\right)\cdot 59^{6} + \left(25 a^{2} + 20 a + 18\right)\cdot 59^{7} + \left(11 a^{2} + 37 a + 50\right)\cdot 59^{8} + \left(3 a^{2} + 7 a + 35\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 27\cdot 59 + 46\cdot 59^{2} + 41\cdot 59^{3} + 12\cdot 59^{4} + 40\cdot 59^{5} + 2\cdot 59^{6} + 12\cdot 59^{7} + 59^{8} + 40\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 49\cdot 59 + 31\cdot 59^{2} + 13\cdot 59^{3} + 39\cdot 59^{4} + 26\cdot 59^{5} + 9\cdot 59^{6} + 5\cdot 59^{7} + 39\cdot 59^{8} + 25\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 46 a + 47 + \left(51 a^{2} + 57 a + 53\right)\cdot 59 + \left(31 a^{2} + 47 a + 5\right)\cdot 59^{2} + \left(14 a^{2} + 15 a\right)\cdot 59^{3} + \left(28 a^{2} + 10 a + 54\right)\cdot 59^{4} + \left(48 a^{2} + 51\right)\cdot 59^{5} + \left(13 a^{2} + 38 a + 30\right)\cdot 59^{6} + \left(26 a^{2} + 36 a + 28\right)\cdot 59^{7} + \left(12 a^{2} + 24 a + 35\right)\cdot 59^{8} + \left(42 a^{2} + 9 a + 34\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 a^{2} + 42 a + 39 + \left(43 a^{2} + 22 a + 48\right)\cdot 59 + \left(26 a^{2} + 15 a + 47\right)\cdot 59^{2} + \left(32 a^{2} + 23 a\right)\cdot 59^{3} + \left(46 a^{2} + 48 a + 56\right)\cdot 59^{4} + \left(43 a^{2} + 11 a + 55\right)\cdot 59^{5} + \left(4 a^{2} + 31 a + 39\right)\cdot 59^{6} + \left(40 a^{2} + 13 a + 15\right)\cdot 59^{7} + \left(26 a^{2} + 47 a + 4\right)\cdot 59^{8} + \left(54 a^{2} + 18 a + 36\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a^{2} + 55 a + 7 + \left(41 a^{2} + 20 a + 52\right)\cdot 59 + \left(50 a^{2} + 36 a + 26\right)\cdot 59^{2} + \left(23 a^{2} + 55 a + 50\right)\cdot 59^{3} + \left(34 a^{2} + 42 a + 58\right)\cdot 59^{4} + \left(43 a^{2} + 31 a + 35\right)\cdot 59^{5} + \left(54 a^{2} + 24 a + 16\right)\cdot 59^{6} + \left(46 a^{2} + 52\right)\cdot 59^{7} + \left(32 a^{2} + 30 a + 42\right)\cdot 59^{8} + \left(49 a^{2} + 30 a + 52\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + 30 a + 44 + \left(23 a^{2} + 37 a + 19\right)\cdot 59 + \left(54 a + 58\right)\cdot 59^{2} + \left(12 a^{2} + 19 a + 30\right)\cdot 59^{3} + \left(43 a^{2} + 5\right)\cdot 59^{4} + \left(25 a^{2} + 47 a + 35\right)\cdot 59^{5} + \left(40 a^{2} + 48 a + 1\right)\cdot 59^{6} + \left(51 a^{2} + 8 a + 15\right)\cdot 59^{7} + \left(19 a^{2} + 46 a + 1\right)\cdot 59^{8} + \left(21 a^{2} + 30 a + 24\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 27 a^{2} + 13 a + 40 + \left(19 a^{2} + 44 a + 57\right)\cdot 59 + \left(10 a^{2} + 42 a + 29\right)\cdot 59^{2} + \left(22 a^{2} + 23 a + 5\right)\cdot 59^{3} + \left(28 a^{2} + 10 a + 19\right)\cdot 59^{4} + \left(41 a^{2} + 26 a + 9\right)\cdot 59^{5} + \left(54 a + 33\right)\cdot 59^{6} + \left(46 a^{2} + 37 a + 29\right)\cdot 59^{7} + \left(14 a^{2} + 50 a + 2\right)\cdot 59^{8} + \left(6 a^{2} + 20 a + 46\right)\cdot 59^{9} +O(59^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $3$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
| $6$ | $2$ | $(3,4)(5,7)$ | $-2$ |
| $6$ | $2$ | $(1,6)(2,8)(3,7)(4,5)$ | $2$ |
| $12$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $-2$ |
| $12$ | $2$ | $(1,8)(5,7)$ | $0$ |
| $32$ | $3$ | $(1,6,8)(3,5,4)$ | $0$ |
| $12$ | $4$ | $(1,4,2,5)(3,8,7,6)$ | $2$ |
| $12$ | $4$ | $(1,5,6,3)(2,4,8,7)$ | $0$ |
| $12$ | $4$ | $(1,6,8,2)(3,7,4,5)$ | $0$ |
| $24$ | $4$ | $(1,7,6,3)(2,4,8,5)$ | $0$ |
| $24$ | $4$ | $(1,8)(3,7,4,5)$ | $0$ |
| $32$ | $6$ | $(1,7)(2,5,6,3,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.