Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(975\)\(\medspace = 3 \cdot 5^{2} \cdot 13 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.1445900625.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.39.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{13})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + x^{6} + 16x^{5} - 7x^{4} - 2x^{3} + 64x^{2} + 8x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 43 + 13\cdot 127 + 7\cdot 127^{2} + 69\cdot 127^{3} + 38\cdot 127^{4} +O(127^{5})\)
$r_{ 2 }$ |
$=$ |
\( 55 + 25\cdot 127 + 98\cdot 127^{2} + 107\cdot 127^{3} + 81\cdot 127^{4} +O(127^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 80 + 105\cdot 127 + 83\cdot 127^{2} + 34\cdot 127^{3} + 14\cdot 127^{4} +O(127^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 102 + 85\cdot 127 + 3\cdot 127^{2} + 74\cdot 127^{3} + 97\cdot 127^{4} +O(127^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 119 + 20\cdot 127 + 13\cdot 127^{2} + 119\cdot 127^{3} + 91\cdot 127^{4} +O(127^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 120 + 74\cdot 127 + 32\cdot 127^{2} + 21\cdot 127^{3} + 11\cdot 127^{4} +O(127^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 121 + 7\cdot 127 + 56\cdot 127^{2} + 93\cdot 127^{3} + 16\cdot 127^{4} +O(127^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 123 + 46\cdot 127 + 86\cdot 127^{2} + 115\cdot 127^{3} + 28\cdot 127^{4} +O(127^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
$2$ | $2$ | $(1,5)(2,4)(3,6)(7,8)$ | $0$ |
$2$ | $4$ | $(1,4,3,7)(2,5,8,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.