Basic invariants
Dimension: | $2$ |
Group: | $D_{8}$ |
Conductor: | \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.190563597.3 |
Galois orbit size: | $2$ |
Smallest permutation container: | $D_{8}$ |
Parity: | odd |
Determinant: | 1.399.2t1.a.a |
Projective image: | $D_4$ |
Projective stem field: | Galois closure of 4.0.1197.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + 2x^{6} - 6x^{5} + 17x^{4} + 3x^{3} - 7x^{2} - x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 12 + 41\cdot 97 + 5\cdot 97^{2} + 60\cdot 97^{3} + 20\cdot 97^{4} + 70\cdot 97^{5} +O(97^{6})\) |
$r_{ 2 }$ | $=$ | \( 14 + 16\cdot 97 + 96\cdot 97^{2} + 29\cdot 97^{3} + 25\cdot 97^{4} + 92\cdot 97^{5} +O(97^{6})\) |
$r_{ 3 }$ | $=$ | \( 15 + 13\cdot 97 + 15\cdot 97^{2} + 53\cdot 97^{3} + 12\cdot 97^{4} + 4\cdot 97^{5} +O(97^{6})\) |
$r_{ 4 }$ | $=$ | \( 39 + 79\cdot 97^{2} + 2\cdot 97^{3} + 89\cdot 97^{4} + 74\cdot 97^{5} +O(97^{6})\) |
$r_{ 5 }$ | $=$ | \( 67 + 48\cdot 97 + 58\cdot 97^{2} + 61\cdot 97^{3} + 39\cdot 97^{4} + 54\cdot 97^{5} +O(97^{6})\) |
$r_{ 6 }$ | $=$ | \( 71 + 19\cdot 97 + 33\cdot 97^{2} + 50\cdot 97^{3} + 93\cdot 97^{4} + 89\cdot 97^{5} +O(97^{6})\) |
$r_{ 7 }$ | $=$ | \( 77 + 64\cdot 97 + 80\cdot 97^{2} + 77\cdot 97^{3} + 79\cdot 97^{4} + 45\cdot 97^{5} +O(97^{6})\) |
$r_{ 8 }$ | $=$ | \( 94 + 86\cdot 97 + 19\cdot 97^{2} + 52\cdot 97^{3} + 27\cdot 97^{4} + 53\cdot 97^{5} +O(97^{6})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $-2$ |
$4$ | $2$ | $(1,2)(4,8)(6,7)$ | $0$ |
$4$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $0$ |
$2$ | $4$ | $(1,4,8,2)(3,6,5,7)$ | $0$ |
$2$ | $8$ | $(1,7,4,3,8,6,2,5)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
$2$ | $8$ | $(1,3,2,7,8,5,4,6)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.