Basic invariants
| Dimension: | $6$ |
| Group: | $A_4^2:C_4$ |
| Conductor: | \(338560000\)\(\medspace = 2^{10} \cdot 5^{4} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.1692800000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_4^2:C_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_4^2:C_4$ |
| Projective stem field: | Galois closure of 8.4.1692800000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + x^{6} - x^{4} - 4x^{3} - 10x^{2} + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 17 a + 2 + \left(8 a^{2} + 17 a\right)\cdot 19 + \left(15 a^{2} + 3 a + 10\right)\cdot 19^{2} + \left(17 a^{2} + 16 a + 6\right)\cdot 19^{3} + \left(12 a^{2} + 17 a + 2\right)\cdot 19^{4} + \left(9 a^{2} + 13 a + 6\right)\cdot 19^{5} + \left(4 a^{2} + 18\right)\cdot 19^{6} + \left(11 a^{2} + 6 a + 4\right)\cdot 19^{7} + \left(14 a + 7\right)\cdot 19^{8} + \left(11 a^{2} + 13 a + 15\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( a^{2} + 2 a + 15 + \left(7 a + 15\right)\cdot 19 + \left(10 a^{2} + 5 a + 1\right)\cdot 19^{2} + \left(8 a^{2} + 11 a + 7\right)\cdot 19^{3} + \left(11 a^{2} + 10 a + 17\right)\cdot 19^{4} + \left(6 a^{2} + 3 a + 16\right)\cdot 19^{5} + \left(12 a^{2} + 18 a + 13\right)\cdot 19^{6} + \left(9 a^{2} + 5 a\right)\cdot 19^{7} + \left(a^{2} + 2 a + 16\right)\cdot 19^{8} + \left(9 a^{2} + 3 a + 3\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a^{2} + 6 a + 12 + \left(18 a^{2} + 14 a + 15\right)\cdot 19 + \left(3 a^{2} + 2 a + 5\right)\cdot 19^{2} + \left(11 a + 4\right)\cdot 19^{3} + \left(11 a^{2} + 8 a + 14\right)\cdot 19^{4} + \left(15 a^{2} + 7 a + 12\right)\cdot 19^{5} + \left(9 a^{2} + 7 a\right)\cdot 19^{6} + \left(10 a^{2} + 2 a + 3\right)\cdot 19^{7} + \left(7 a^{2} + a + 2\right)\cdot 19^{8} + \left(10 a^{2} + 6 a + 12\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 15 a + 6 + 17\cdot 19 + \left(8 a^{2} + 9 a + 3\right)\cdot 19^{2} + \left(7 a^{2} + 18 a + 17\right)\cdot 19^{3} + \left(6 a^{2} + 10 a + 1\right)\cdot 19^{4} + \left(15 a^{2} + 3 a + 12\right)\cdot 19^{5} + \left(11 a^{2} + 3 a + 18\right)\cdot 19^{6} + \left(7 a^{2} + 17 a + 7\right)\cdot 19^{7} + \left(17 a + 8\right)\cdot 19^{8} + \left(9 a^{2} + 12 a + 8\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 + 14\cdot 19 + 18\cdot 19^{2} + 18\cdot 19^{3} + 16\cdot 19^{4} + 9\cdot 19^{5} + 11\cdot 19^{6} + 4\cdot 19^{7} + 13\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 + 3\cdot 19 + 13\cdot 19^{2} + 16\cdot 19^{3} + 10\cdot 19^{4} + 2\cdot 19^{5} + 8\cdot 19^{6} + 12\cdot 19^{7} + 17\cdot 19^{8} + 18\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 a^{2} + 1 + \left(10 a^{2} + 13 a + 12\right)\cdot 19 + \left(12 a^{2} + 9 a + 8\right)\cdot 19^{2} + \left(11 a^{2} + 10 a + 15\right)\cdot 19^{3} + \left(13 a^{2} + 9 a + 16\right)\cdot 19^{4} + \left(2 a^{2} + a + 12\right)\cdot 19^{5} + \left(2 a^{2} + 5\right)\cdot 19^{6} + \left(17 a^{2} + 7 a + 14\right)\cdot 19^{7} + \left(16 a^{2} + 2 a + 12\right)\cdot 19^{8} + \left(17 a^{2} + 2 a + 14\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 17 a + 9 + \left(3 a + 16\right)\cdot 19 + \left(7 a^{2} + 7 a + 13\right)\cdot 19^{2} + \left(11 a^{2} + 8 a + 8\right)\cdot 19^{3} + \left(a^{2} + 18 a + 14\right)\cdot 19^{4} + \left(7 a^{2} + 7 a + 2\right)\cdot 19^{5} + \left(16 a^{2} + 8 a + 18\right)\cdot 19^{6} + \left(18 a + 8\right)\cdot 19^{7} + \left(11 a^{2} + 18 a + 11\right)\cdot 19^{8} + \left(18 a^{2} + 18 a + 8\right)\cdot 19^{9} +O(19^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $6$ | $2$ | $(3,6)(4,8)$ | $2$ |
| $9$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-2$ |
| $36$ | $2$ | $(2,7)(4,6)$ | $2$ |
| $16$ | $3$ | $(3,4,6)$ | $3$ |
| $64$ | $3$ | $(2,7,5)(3,4,6)$ | $0$ |
| $36$ | $4$ | $(1,7,5,2)(3,6,8,4)$ | $-2$ |
| $72$ | $4$ | $(3,4,6,8)(5,7)$ | $0$ |
| $72$ | $4$ | $(1,8)(2,4,7,6)(3,5)$ | $0$ |
| $72$ | $4$ | $(1,8)(2,6,7,4)(3,5)$ | $0$ |
| $48$ | $6$ | $(1,5)(2,7)(3,6,8)$ | $-1$ |
| $72$ | $8$ | $(1,6,7,8,5,4,2,3)$ | $0$ |
| $72$ | $8$ | $(1,8,2,6,5,3,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.