Basic invariants
Dimension: | $4$ |
Group: | $C_2 \wr C_2\wr C_2$ |
Conductor: | \(12753\)\(\medspace = 3^{2} \cdot 13 \cdot 109 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.1492101.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2 \wr C_2\wr C_2$ |
Parity: | even |
Determinant: | 1.1417.2t1.a.a |
Projective image: | $C_2\wr C_2^2$ |
Projective stem field: | Galois closure of 8.4.3053999169.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + x^{6} - 3x^{5} + x^{4} - 2x^{3} + 3x^{2} + 1 \) . |
The roots of $f$ are computed in $\Q_{ 823 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 39 + 731\cdot 823 + 15\cdot 823^{2} + 286\cdot 823^{3} + 156\cdot 823^{4} + 761\cdot 823^{5} +O(823^{6})\)
$r_{ 2 }$ |
$=$ |
\( 126 + 533\cdot 823 + 352\cdot 823^{2} + 551\cdot 823^{3} + 586\cdot 823^{4} + 88\cdot 823^{5} +O(823^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 157 + 531\cdot 823 + 205\cdot 823^{2} + 500\cdot 823^{3} + 474\cdot 823^{4} + 570\cdot 823^{5} +O(823^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 227 + 761\cdot 823 + 375\cdot 823^{2} + 408\cdot 823^{3} + 817\cdot 823^{4} + 122\cdot 823^{5} +O(823^{6})\)
| $r_{ 5 }$ |
$=$ |
\( 338 + 423\cdot 823 + 58\cdot 823^{2} + 525\cdot 823^{3} + 692\cdot 823^{4} + 351\cdot 823^{5} +O(823^{6})\)
| $r_{ 6 }$ |
$=$ |
\( 413 + 815\cdot 823 + 82\cdot 823^{2} + 389\cdot 823^{3} + 9\cdot 823^{4} + 626\cdot 823^{5} +O(823^{6})\)
| $r_{ 7 }$ |
$=$ |
\( 464 + 698\cdot 823 + 420\cdot 823^{2} + 810\cdot 823^{3} + 362\cdot 823^{4} + 746\cdot 823^{5} +O(823^{6})\)
| $r_{ 8 }$ |
$=$ |
\( 706 + 443\cdot 823 + 133\cdot 823^{2} + 644\cdot 823^{3} + 191\cdot 823^{4} + 24\cdot 823^{5} +O(823^{6})\)
| |
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 |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $-4$ |
$2$ | $2$ | $(1,3)(5,7)$ | $0$ |
$4$ | $2$ | $(1,3)$ | $2$ |
$4$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
$4$ | $2$ | $(1,3)(2,8)$ | $0$ |
$4$ | $2$ | $(1,3)(2,8)(4,6)$ | $-2$ |
$4$ | $2$ | $(1,3)(2,6)(4,8)(5,7)$ | $-2$ |
$4$ | $2$ | $(1,5)(3,7)$ | $2$ |
$8$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $0$ |
$8$ | $2$ | $(2,6)(4,8)(5,7)$ | $0$ |
$4$ | $4$ | $(1,7,3,5)(2,6,8,4)$ | $0$ |
$4$ | $4$ | $(1,5,3,7)$ | $2$ |
$4$ | $4$ | $(1,7,3,5)(2,8)(4,6)$ | $-2$ |
$8$ | $4$ | $(1,8,3,2)(4,5,6,7)$ | $0$ |
$8$ | $4$ | $(1,5,3,7)(2,8)$ | $0$ |
$8$ | $4$ | $(1,5)(2,6,8,4)(3,7)$ | $0$ |
$16$ | $4$ | $(1,4,7,2)(3,6,5,8)$ | $0$ |
$16$ | $4$ | $(1,8)(2,3)(4,5,6,7)$ | $0$ |
$16$ | $8$ | $(1,4,7,2,3,6,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.