Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.25344958401.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.399.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{133})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} + 11x^{6} + 124x^{4} - 33x^{2} + 9 \) . |
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 49 + 116\cdot 223 + 74\cdot 223^{2} + 150\cdot 223^{3} + 169\cdot 223^{4} +O(223^{5})\)
$r_{ 2 }$ |
$=$ |
\( 53 + 111\cdot 223 + 101\cdot 223^{2} + 223^{3} + 89\cdot 223^{4} +O(223^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 96 + 112\cdot 223 + 157\cdot 223^{2} + 128\cdot 223^{3} + 70\cdot 223^{4} +O(223^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 110 + 201\cdot 223 + 112\cdot 223^{2} + 59\cdot 223^{3} + 103\cdot 223^{4} +O(223^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 113 + 21\cdot 223 + 110\cdot 223^{2} + 163\cdot 223^{3} + 119\cdot 223^{4} +O(223^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 127 + 110\cdot 223 + 65\cdot 223^{2} + 94\cdot 223^{3} + 152\cdot 223^{4} +O(223^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 170 + 111\cdot 223 + 121\cdot 223^{2} + 221\cdot 223^{3} + 133\cdot 223^{4} +O(223^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 174 + 106\cdot 223 + 148\cdot 223^{2} + 72\cdot 223^{3} + 53\cdot 223^{4} +O(223^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
$2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
$2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.