Basic invariants
| Dimension: | $4$ |
| Group: | $C_2 \wr C_2\wr C_2$ |
| Conductor: | \(1917981\)\(\medspace = 3^{2} \cdot 13^{3} \cdot 97 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1327833.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2 \wr C_2\wr C_2$ |
| Parity: | even |
| Determinant: | 1.1261.2t1.a.a |
| Projective image: | $C_2\wr C_2^2$ |
| Projective stem field: | Galois closure of 8.4.2418574041.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} + x^{6} - 2x^{5} + 3x^{4} - 4x^{3} + 4x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + \left(11 a + 7\right)\cdot 19 + 9 a\cdot 19^{2} + 8 a\cdot 19^{3} + \left(8 a + 16\right)\cdot 19^{4} + \left(15 a + 8\right)\cdot 19^{5} + \left(4 a + 1\right)\cdot 19^{6} + \left(14 a + 9\right)\cdot 19^{7} + \left(6 a + 14\right)\cdot 19^{8} + \left(18 a + 8\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 13 + \left(7 a + 5\right)\cdot 19 + \left(9 a + 17\right)\cdot 19^{2} + \left(10 a + 17\right)\cdot 19^{3} + \left(10 a + 15\right)\cdot 19^{4} + \left(3 a + 15\right)\cdot 19^{5} + \left(14 a + 9\right)\cdot 19^{6} + \left(4 a + 18\right)\cdot 19^{7} + \left(12 a + 6\right)\cdot 19^{8} + 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 11\cdot 19 + 17\cdot 19^{2} + 6\cdot 19^{3} + 16\cdot 19^{4} + 9\cdot 19^{5} + 9\cdot 19^{6} + 7\cdot 19^{7} + 8\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + \left(12 a + 17\right)\cdot 19 + \left(6 a + 18\right)\cdot 19^{2} + \left(10 a + 7\right)\cdot 19^{3} + \left(a + 16\right)\cdot 19^{4} + \left(8 a + 3\right)\cdot 19^{5} + \left(2 a + 6\right)\cdot 19^{6} + \left(3 a + 12\right)\cdot 19^{7} + \left(5 a + 1\right)\cdot 19^{8} + \left(9 a + 9\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 18 + \left(6 a + 11\right)\cdot 19 + \left(12 a + 12\right)\cdot 19^{2} + \left(8 a + 11\right)\cdot 19^{3} + \left(17 a + 7\right)\cdot 19^{4} + \left(10 a + 10\right)\cdot 19^{5} + 16 a\cdot 19^{6} + \left(15 a + 13\right)\cdot 19^{7} + \left(13 a + 3\right)\cdot 19^{8} + \left(9 a + 13\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + \left(16 a + 10\right)\cdot 19 + \left(18 a + 12\right)\cdot 19^{2} + \left(12 a + 4\right)\cdot 19^{3} + \left(2 a + 9\right)\cdot 19^{4} + \left(13 a + 17\right)\cdot 19^{5} + \left(8 a + 10\right)\cdot 19^{6} + \left(10 a + 15\right)\cdot 19^{7} + \left(4 a + 13\right)\cdot 19^{8} + \left(6 a + 17\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 2 a + 17 + \left(2 a + 9\right)\cdot 19 + 14\cdot 19^{2} + \left(6 a + 17\right)\cdot 19^{3} + \left(16 a + 17\right)\cdot 19^{4} + \left(5 a + 8\right)\cdot 19^{5} + \left(10 a + 6\right)\cdot 19^{6} + \left(8 a + 17\right)\cdot 19^{7} + \left(14 a + 7\right)\cdot 19^{8} + 12 a\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 11 + 2\cdot 19 + 19^{2} + 9\cdot 19^{3} + 14\cdot 19^{4} + 12\cdot 19^{6} + 19^{7} + 8\cdot 19^{8} + 17\cdot 19^{9} +O(19^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,4)(3,8)(6,7)$ | $-4$ |
| $2$ | $2$ | $(3,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(3,8)$ | $2$ |
| $4$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(3,8)$ | $0$ |
| $4$ | $2$ | $(3,6)(7,8)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,8)(4,5)(6,7)$ | $2$ |
| $4$ | $2$ | $(1,5)(3,8)(6,7)$ | $-2$ |
| $8$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $8$ | $2$ | $(1,2)(3,8)(4,5)$ | $0$ |
| $4$ | $4$ | $(1,4,5,2)(3,7,8,6)$ | $0$ |
| $4$ | $4$ | $(3,7,8,6)$ | $-2$ |
| $4$ | $4$ | $(1,4,5,2)(3,8)(6,7)$ | $2$ |
| $8$ | $4$ | $(1,3,5,8)(2,6,4,7)$ | $0$ |
| $8$ | $4$ | $(1,2)(3,7,8,6)(4,5)$ | $0$ |
| $8$ | $4$ | $(1,5)(3,7,8,6)$ | $0$ |
| $16$ | $4$ | $(1,3,2,6)(4,7,5,8)$ | $0$ |
| $16$ | $4$ | $(1,3,5,8)(2,6)(4,7)$ | $0$ |
| $16$ | $8$ | $(1,3,4,7,5,8,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.