Basic invariants
Dimension: | $2$ |
Group: | $C_8:C_2$ |
Conductor: | \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \) |
Artin stem field: | Galois closure of 8.8.414720000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_8:C_2$ |
Parity: | even |
Determinant: | 1.40.4t1.a.b |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{5})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 30x^{6} + 225x^{4} - 540x^{2} + 405 \) . |
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.
Roots:
$r_{ 1 }$ | $=$ |
\( 5 + 13\cdot 41 + 27\cdot 41^{2} + 10\cdot 41^{3} + 41^{4} + 15\cdot 41^{5} + 20\cdot 41^{7} + 28\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\)
$r_{ 2 }$ |
$=$ |
\( 7 + 41 + 27\cdot 41^{2} + 9\cdot 41^{3} + 27\cdot 41^{4} + 19\cdot 41^{5} + 24\cdot 41^{6} + 36\cdot 41^{7} + 23\cdot 41^{8} + 25\cdot 41^{9} +O(41^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 13 + 17\cdot 41 + 32\cdot 41^{2} + 9\cdot 41^{3} + 10\cdot 41^{4} + 20\cdot 41^{5} + 35\cdot 41^{6} + 5\cdot 41^{7} + 31\cdot 41^{8} + 16\cdot 41^{9} +O(41^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 19 + 36\cdot 41 + 41^{2} + 40\cdot 41^{3} + 3\cdot 41^{4} + 33\cdot 41^{5} + 30\cdot 41^{6} + 5\cdot 41^{7} + 40\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 22 + 4\cdot 41 + 39\cdot 41^{2} + 37\cdot 41^{4} + 7\cdot 41^{5} + 10\cdot 41^{6} + 35\cdot 41^{7} + 26\cdot 41^{9} +O(41^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 28 + 23\cdot 41 + 8\cdot 41^{2} + 31\cdot 41^{3} + 30\cdot 41^{4} + 20\cdot 41^{5} + 5\cdot 41^{6} + 35\cdot 41^{7} + 9\cdot 41^{8} + 24\cdot 41^{9} +O(41^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 34 + 39\cdot 41 + 13\cdot 41^{2} + 31\cdot 41^{3} + 13\cdot 41^{4} + 21\cdot 41^{5} + 16\cdot 41^{6} + 4\cdot 41^{7} + 17\cdot 41^{8} + 15\cdot 41^{9} +O(41^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 36 + 27\cdot 41 + 13\cdot 41^{2} + 30\cdot 41^{3} + 39\cdot 41^{4} + 25\cdot 41^{5} + 40\cdot 41^{6} + 20\cdot 41^{7} + 12\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(1,8)(4,5)$ | $0$ |
$1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-2 \zeta_{4}$ |
$1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $2 \zeta_{4}$ |
$2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
$2$ | $8$ | $(1,6,4,7,8,3,5,2)$ | $0$ |
$2$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $0$ |
$2$ | $8$ | $(1,3,5,7,8,6,4,2)$ | $0$ |
$2$ | $8$ | $(1,7,4,3,8,2,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.