Basic invariants
Dimension: | $6$ |
Group: | $C_2^4:C_6$ |
Conductor: | \(848081520125\)\(\medspace = 5^{3} \cdot 7^{4} \cdot 41^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.4.2522550625.1 |
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 8.4.2522550625.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{7} - x^{6} + 9x^{5} - 11x^{4} + 11x^{3} - 13x^{2} + 21x - 9 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 2 + 11 + 3\cdot 11^{2} + 9\cdot 11^{3} + 2\cdot 11^{4} + 5\cdot 11^{6} + 3\cdot 11^{7} + 3\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\)
$r_{ 2 }$ |
$=$ |
\( 8 a^{2} + 5 a + 8 + \left(8 a^{2} + 5 a + 9\right)\cdot 11 + \left(8 a^{2} + 8 a + 9\right)\cdot 11^{2} + \left(5 a + 4\right)\cdot 11^{3} + \left(6 a^{2} + 9 a + 9\right)\cdot 11^{4} + \left(5 a^{2} + 1\right)\cdot 11^{5} + \left(5 a^{2} + 2 a + 10\right)\cdot 11^{6} + \left(a^{2} + a + 6\right)\cdot 11^{7} + \left(2 a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(4 a^{2} + 7 a\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 2 a^{2} + 7 a + \left(2 a^{2} + 3 a\right)\cdot 11 + \left(9 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(8 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(9 a^{2} + 6 a + 10\right)\cdot 11^{4} + \left(10 a^{2} + 1\right)\cdot 11^{5} + \left(5 a^{2} + 7 a + 7\right)\cdot 11^{6} + \left(5 a^{2} + 10 a + 6\right)\cdot 11^{7} + \left(8 a^{2} + 2 a + 1\right)\cdot 11^{8} + \left(3 a^{2} + 4 a + 4\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 3 a + 1 + \left(6 a^{2} + 10 a + 6\right)\cdot 11 + \left(5 a^{2} + 8 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 7 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 4 a + 4\right)\cdot 11^{4} + \left(4 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{6} + \left(4 a^{2} + 3 a\right)\cdot 11^{7} + \left(7 a^{2} + a + 5\right)\cdot 11^{8} + \left(6 a^{2} + 8 a + 3\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a^{2} + a + 5 + \left(7 a^{2} + 5 a + 10\right)\cdot 11 + \left(9 a^{2} + 7 a\right)\cdot 11^{2} + \left(6 a^{2} + 10 a + 7\right)\cdot 11^{3} + \left(10 a^{2} + 7 a + 7\right)\cdot 11^{4} + \left(3 a^{2} + 5 a + 3\right)\cdot 11^{5} + \left(2 a^{2} + 3 a + 2\right)\cdot 11^{6} + \left(a^{2} + 4 a + 8\right)\cdot 11^{7} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{8} + \left(10 a^{2} + 4 a + 9\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 3 a^{2} + 3 a + 5 + \left(7 a^{2} + 6 a\right)\cdot 11 + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{2} + \left(7 a^{2} + 8 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 7 a + 1\right)\cdot 11^{4} + \left(a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(9 a^{2} + 4 a + 7\right)\cdot 11^{6} + \left(4 a^{2} + 6 a + 7\right)\cdot 11^{7} + \left(a^{2} + 2 a\right)\cdot 11^{8} + \left(6 a + 2\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 6 + 2\cdot 11 + 5\cdot 11^{2} + 7\cdot 11^{3} + 11^{4} + 10\cdot 11^{5} + 10\cdot 11^{6} + 5\cdot 11^{7} + 7\cdot 11^{8} + 3\cdot 11^{9} +O(11^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 6 a^{2} + 3 a + 9 + \left(a^{2} + 2 a + 2\right)\cdot 11 + \left(3 a^{2} + a + 3\right)\cdot 11^{2} + \left(6 a^{2} + 7 a + 6\right)\cdot 11^{3} + \left(a^{2} + 7 a + 6\right)\cdot 11^{4} + \left(7 a^{2} + 4 a\right)\cdot 11^{5} + \left(2 a^{2} + 10\right)\cdot 11^{6} + \left(4 a^{2} + 7 a + 4\right)\cdot 11^{7} + \left(4 a^{2} + 5 a + 3\right)\cdot 11^{8} + \left(7 a^{2} + 2 a + 5\right)\cdot 11^{9} +O(11^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $6$ |
$3$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $-2$ |
$4$ | $2$ | $(1,6)(2,3)(4,8)(5,7)$ | $0$ |
$6$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $2$ |
$6$ | $2$ | $(2,6)(4,7)$ | $-2$ |
$16$ | $3$ | $(1,8,3)(2,4,7)$ | $0$ |
$16$ | $3$ | $(1,3,8)(2,7,4)$ | $0$ |
$12$ | $4$ | $(1,4,3,7)(2,8,6,5)$ | $0$ |
$16$ | $6$ | $(1,7,8,6,5,4)(2,3)$ | $0$ |
$16$ | $6$ | $(1,4,5,6,8,7)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.