Basic invariants
Dimension: | $2$ |
Group: | $D_4$ |
Conductor: | \(363\)\(\medspace = 3 \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin field: | Galois closure of 8.0.143496441.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-11})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + 2x^{6} - 3x^{5} - x^{4} + 6x^{3} + 8x^{2} + 8x + 16 \) . |
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 7 + 81\cdot 97 + 35\cdot 97^{2} + 85\cdot 97^{3} + 49\cdot 97^{4} +O(97^{5})\)
$r_{ 2 }$ |
$=$ |
\( 8 + 66\cdot 97 + 24\cdot 97^{2} + 15\cdot 97^{3} + 37\cdot 97^{4} +O(97^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 9 + 35\cdot 97 + 21\cdot 97^{2} + 41\cdot 97^{3} + 90\cdot 97^{4} +O(97^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 24 + 20\cdot 97 + 76\cdot 97^{2} + 65\cdot 97^{3} + 49\cdot 97^{4} +O(97^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 38 + 59\cdot 97 + 21\cdot 97^{2} + 3\cdot 97^{3} + 82\cdot 97^{4} +O(97^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 51 + 76\cdot 97 + 36\cdot 97^{2} + 57\cdot 97^{3} + 70\cdot 97^{4} +O(97^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 69 + 60\cdot 97 + 5\cdot 97^{2} + 96\cdot 97^{3} + 78\cdot 97^{4} +O(97^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 86 + 85\cdot 97 + 68\cdot 97^{2} + 23\cdot 97^{3} + 26\cdot 97^{4} +O(97^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
$2$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $0$ |
$2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
$2$ | $4$ | $(1,2,5,3)(4,6,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.