Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(111\)\(\medspace = 3 \cdot 37 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.151807041.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.111.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 5x^{6} + 28x^{4} + 15x^{2} + 9 \) . |
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 16 + 101\cdot 127 + 33\cdot 127^{2} + 28\cdot 127^{3} + 49\cdot 127^{4} +O(127^{5})\)
$r_{ 2 }$ |
$=$ |
\( 41 + 37\cdot 127 + 52\cdot 127^{2} + 77\cdot 127^{3} + 123\cdot 127^{4} +O(127^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 58 + 59\cdot 127 + 59\cdot 127^{2} + 68\cdot 127^{3} + 88\cdot 127^{4} +O(127^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 61 + 20\cdot 127 + 112\cdot 127^{2} + 110\cdot 127^{3} + 116\cdot 127^{4} +O(127^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 66 + 106\cdot 127 + 14\cdot 127^{2} + 16\cdot 127^{3} + 10\cdot 127^{4} +O(127^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 69 + 67\cdot 127 + 67\cdot 127^{2} + 58\cdot 127^{3} + 38\cdot 127^{4} +O(127^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 86 + 89\cdot 127 + 74\cdot 127^{2} + 49\cdot 127^{3} + 3\cdot 127^{4} +O(127^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 111 + 25\cdot 127 + 93\cdot 127^{2} + 98\cdot 127^{3} + 77\cdot 127^{4} +O(127^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
$2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
$2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.