Basic invariants
| Dimension: | $6$ |
| Group: | $C_2^4:C_6$ |
| Conductor: | \(252405125\)\(\medspace = 5^{3} \cdot 7^{4} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.1262025625.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2^4:C_6$ |
| Parity: | even |
| Determinant: | 1.5.2t1.a.a |
| Projective image: | $C_2^3:A_4$ |
| Projective stem field: | Galois closure of 8.0.1262025625.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 4x^{5} + 12x^{4} + 2x^{3} - 14x^{2} - 5x + 11 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{3} + 2x + 9 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 9\cdot 11 + 10\cdot 11^{2} + 8\cdot 11^{3} + 6\cdot 11^{4} + 5\cdot 11^{5} + 7\cdot 11^{6} + 10\cdot 11^{7} + 3\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a^{2} + 4 a + 10 + \left(3 a^{2} + 6 a + 3\right)\cdot 11 + \left(8 a^{2} + 6 a + 10\right)\cdot 11^{2} + \left(9 a + 3\right)\cdot 11^{3} + \left(3 a^{2} + a\right)\cdot 11^{4} + \left(a^{2} + 10 a + 3\right)\cdot 11^{5} + \left(9 a^{2} + 2 a + 2\right)\cdot 11^{6} + \left(2 a^{2} + 2 a + 4\right)\cdot 11^{7} + \left(2 a^{2} + 6 a + 1\right)\cdot 11^{8} + \left(2 a^{2} + 8 a + 5\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 1 + \left(2 a + 3\right)\cdot 11 + \left(a^{2} + a + 4\right)\cdot 11^{2} + \left(10 a + 10\right)\cdot 11^{3} + \left(4 a^{2} + 2 a + 8\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(5 a^{2} + 6 a + 1\right)\cdot 11^{6} + \left(9 a^{2} + 2\right)\cdot 11^{7} + \left(4 a^{2} + 6 a + 1\right)\cdot 11^{8} + \left(6 a^{2} + 7 a + 7\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a^{2} + \left(8 a^{2} + 3 a + 5\right)\cdot 11 + \left(3 a^{2} + 8 a + 1\right)\cdot 11^{2} + \left(3 a^{2} + 5 a + 5\right)\cdot 11^{3} + \left(9 a^{2} + 5 a + 6\right)\cdot 11^{4} + \left(3 a + 6\right)\cdot 11^{5} + \left(4 a^{2} + 6 a + 6\right)\cdot 11^{6} + \left(3 a^{2} + 8 a + 4\right)\cdot 11^{7} + \left(10 a^{2} + 4 a + 1\right)\cdot 11^{8} + \left(6 a^{2} + 4 a + 4\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a^{2} + 2 a + \left(2 a^{2} + 5 a + 8\right)\cdot 11 + \left(7 a^{2} + 5 a + 9\right)\cdot 11^{2} + \left(4 a^{2} + 6\right)\cdot 11^{3} + \left(6 a^{2} + 10 a + 2\right)\cdot 11^{4} + \left(9 a^{2} + 4 a + 7\right)\cdot 11^{5} + \left(8 a^{2} + a + 5\right)\cdot 11^{6} + \left(5 a^{2} + 8 a\right)\cdot 11^{7} + \left(6 a^{2} + 10 a\right)\cdot 11^{8} + \left(3 a^{2} + 8 a + 7\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 a^{2} + 3 a + 3 + \left(7 a^{2} + 2 a + 2\right)\cdot 11 + \left(a^{2} + 3 a + 5\right)\cdot 11^{2} + \left(10 a^{2} + 2 a + 5\right)\cdot 11^{3} + \left(3 a^{2} + 6 a + 1\right)\cdot 11^{4} + \left(10 a^{2} + 8 a + 4\right)\cdot 11^{5} + \left(6 a^{2} + a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 8 a + 5\right)\cdot 11^{7} + \left(3 a^{2} + 9 a + 3\right)\cdot 11^{8} + \left(2 a^{2} + 5 a + 5\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 a^{2} + 9 a + 1 + \left(10 a^{2} + 2 a\right)\cdot 11 + \left(10 a^{2} + 8 a\right)\cdot 11^{2} + \left(2 a^{2} + 4 a + 1\right)\cdot 11^{3} + \left(6 a^{2} + 6 a + 6\right)\cdot 11^{4} + \left(2 a + 2\right)\cdot 11^{5} + \left(9 a^{2} + 3 a + 2\right)\cdot 11^{6} + \left(a^{2} + 5 a + 6\right)\cdot 11^{7} + \left(5 a^{2} + 6 a + 5\right)\cdot 11^{8} + \left(8 a + 6\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 + 11 + 2\cdot 11^{2} + 2\cdot 11^{3} + 7\cdot 11^{5} + 7\cdot 11^{6} + 9\cdot 11^{7} + 4\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $3$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ |
| $6$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $-2$ |
| $6$ | $2$ | $(2,8)(3,6)$ | $2$ |
| $16$ | $3$ | $(1,4,5)(2,6,3)$ | $0$ |
| $16$ | $3$ | $(1,5,4)(2,3,6)$ | $0$ |
| $12$ | $4$ | $(1,8,5,3)(2,7,6,4)$ | $0$ |
| $16$ | $6$ | $(1,3,7,8,5,6)(2,4)$ | $0$ |
| $16$ | $6$ | $(1,6,5,8,7,3)(2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.