Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr C_2\wr C_2$ |
| Conductor: | \(46518691401\)\(\medspace = 3^{2} \cdot 7^{3} \cdot 13^{3} \cdot 19^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1820637.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr C_2\wr C_2$ |
| Parity: | even |
| Determinant: | 1.1729.2t1.a.a |
| Projective image: | $C_2\wr C_2^2$ |
| Projective stem field: | Galois closure of 8.4.4546939761.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x^{5} - x^{4} - 2x^{3} + 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$:
\( x^{2} + 63x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 7 + \left(18 a + 64\right)\cdot 67 + \left(38 a + 1\right)\cdot 67^{2} + \left(55 a + 40\right)\cdot 67^{3} + \left(60 a + 41\right)\cdot 67^{4} + 58\cdot 67^{5} + \left(13 a + 46\right)\cdot 67^{6} + \left(26 a + 62\right)\cdot 67^{7} + \left(24 a + 49\right)\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 32 + 3\cdot 67 + 48\cdot 67^{2} + 59\cdot 67^{3} + 10\cdot 67^{4} + 12\cdot 67^{5} + 46\cdot 67^{6} + 5\cdot 67^{7} + 16\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 40 a + 6 + \left(59 a + 51\right)\cdot 67 + \left(36 a + 20\right)\cdot 67^{2} + \left(29 a + 32\right)\cdot 67^{3} + \left(15 a + 54\right)\cdot 67^{4} + \left(52 a + 27\right)\cdot 67^{5} + \left(33 a + 17\right)\cdot 67^{6} + \left(49 a + 23\right)\cdot 67^{7} + \left(44 a + 63\right)\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a + 32 + \left(7 a + 48\right)\cdot 67 + \left(30 a + 41\right)\cdot 67^{2} + \left(37 a + 46\right)\cdot 67^{3} + \left(51 a + 19\right)\cdot 67^{4} + \left(14 a + 20\right)\cdot 67^{5} + \left(33 a + 33\right)\cdot 67^{6} + \left(17 a + 53\right)\cdot 67^{7} + \left(22 a + 58\right)\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 + 59\cdot 67^{2} + 11\cdot 67^{3} + 11\cdot 67^{4} + 23\cdot 67^{5} + 54\cdot 67^{6} + 26\cdot 67^{7} + 7\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 a + 63 + \left(48 a + 55\right)\cdot 67 + \left(28 a + 2\right)\cdot 67^{2} + \left(11 a + 23\right)\cdot 67^{3} + \left(6 a + 28\right)\cdot 67^{4} + \left(66 a + 1\right)\cdot 67^{5} + \left(53 a + 31\right)\cdot 67^{6} + \left(40 a + 20\right)\cdot 67^{7} + \left(42 a + 54\right)\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 48 a + 34 + \left(36 a + 6\right)\cdot 67 + \left(28 a + 8\right)\cdot 67^{2} + \left(28 a + 18\right)\cdot 67^{3} + \left(13 a + 38\right)\cdot 67^{4} + \left(65 a + 5\right)\cdot 67^{5} + \left(16 a + 18\right)\cdot 67^{6} + \left(43 a + 60\right)\cdot 67^{7} + \left(60 a + 9\right)\cdot 67^{8} +O(67^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 19 a + 25 + \left(30 a + 38\right)\cdot 67 + \left(38 a + 18\right)\cdot 67^{2} + \left(38 a + 36\right)\cdot 67^{3} + \left(53 a + 63\right)\cdot 67^{4} + \left(a + 51\right)\cdot 67^{5} + \left(50 a + 20\right)\cdot 67^{6} + \left(23 a + 15\right)\cdot 67^{7} + \left(6 a + 8\right)\cdot 67^{8} +O(67^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-4$ |
| $2$ | $2$ | $(3,8)(4,7)$ | $0$ |
| $4$ | $2$ | $(2,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(3,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,5)(3,7)(4,8)$ | $2$ |
| $4$ | $2$ | $(3,7)(4,8)$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,5)(3,8)$ | $2$ |
| $8$ | $2$ | $(2,5)(3,7)(4,8)$ | $0$ |
| $8$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $4$ | $4$ | $(1,5,6,2)(3,4,8,7)$ | $0$ |
| $4$ | $4$ | $(3,4,8,7)$ | $2$ |
| $4$ | $4$ | $(1,5,6,2)(3,8)(4,7)$ | $-2$ |
| $8$ | $4$ | $(1,3,6,8)(2,4,5,7)$ | $0$ |
| $8$ | $4$ | $(1,6)(3,4,8,7)$ | $0$ |
| $8$ | $4$ | $(1,5,6,2)(3,4)(7,8)$ | $0$ |
| $16$ | $4$ | $(1,3)(2,4,5,7)(6,8)$ | $0$ |
| $16$ | $4$ | $(1,3,2,7)(4,6,8,5)$ | $0$ |
| $16$ | $8$ | $(1,3,5,4,6,8,2,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.