Basic invariants
Dimension: | $2$ |
Group: | $C_8:C_2$ |
Conductor: | \(9225\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 41 \) |
Artin stem field: | Galois closure of 8.4.95738203125.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_8:C_2$ |
Parity: | odd |
Determinant: | 1.205.4t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{-123})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{7} - 2x^{6} + 9x^{5} - 90x^{4} + 84x^{3} + 208x^{2} + 327x + 211 \) . |
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 29\cdot 31^{2} + 2\cdot 31^{3} + 25\cdot 31^{4} +O(31^{5})\)
$r_{ 2 }$ |
$=$ |
\( 5 + 22\cdot 31 + 6\cdot 31^{2} + 25\cdot 31^{3} + 7\cdot 31^{4} +O(31^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 7 + 15\cdot 31 + 17\cdot 31^{2} + 20\cdot 31^{3} + 22\cdot 31^{4} +O(31^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 11 + 16\cdot 31 + 17\cdot 31^{2} + 3\cdot 31^{3} + 28\cdot 31^{4} +O(31^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 12 + 14\cdot 31 + 23\cdot 31^{2} + 10\cdot 31^{3} + 16\cdot 31^{4} +O(31^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 17 + 15\cdot 31 + 26\cdot 31^{2} + 22\cdot 31^{3} + 8\cdot 31^{4} +O(31^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 18 + 9\cdot 31 + 2\cdot 31^{2} + 2\cdot 31^{3} + 20\cdot 31^{4} +O(31^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 24 + 30\cdot 31 + 5\cdot 31^{3} + 26\cdot 31^{4} +O(31^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $-2$ |
$2$ | $2$ | $(1,8)(3,7)$ | $0$ |
$1$ | $4$ | $(1,7,8,3)(2,5,4,6)$ | $2 \zeta_{4}$ |
$1$ | $4$ | $(1,3,8,7)(2,6,4,5)$ | $-2 \zeta_{4}$ |
$2$ | $4$ | $(1,7,8,3)(2,6,4,5)$ | $0$ |
$2$ | $8$ | $(1,2,7,5,8,4,3,6)$ | $0$ |
$2$ | $8$ | $(1,5,3,2,8,6,7,4)$ | $0$ |
$2$ | $8$ | $(1,4,3,5,8,2,7,6)$ | $0$ |
$2$ | $8$ | $(1,5,7,4,8,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.