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 8.0.414720000000.3 |
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:
$r_{ 1 }$ | $=$ | \( 4 + 40\cdot 41 + 37\cdot 41^{3} + 9\cdot 41^{4} + 8\cdot 41^{5} + 4\cdot 41^{6} + 27\cdot 41^{7} + 24\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\) |
$r_{ 2 }$ | $=$ | \( 6 + 14\cdot 41 + 12\cdot 41^{2} + 16\cdot 41^{3} + 33\cdot 41^{4} + 12\cdot 41^{5} + 38\cdot 41^{6} + 34\cdot 41^{7} + 19\cdot 41^{8} + 17\cdot 41^{9} +O(41^{10})\) |
$r_{ 3 }$ | $=$ | \( 7 + 7\cdot 41 + 7\cdot 41^{2} + 26\cdot 41^{3} + 37\cdot 41^{4} + 39\cdot 41^{5} + 17\cdot 41^{6} + 29\cdot 41^{7} + 38\cdot 41^{8} + 17\cdot 41^{9} +O(41^{10})\) |
$r_{ 4 }$ | $=$ | \( 19 + 8\cdot 41 + 9\cdot 41^{3} + 31\cdot 41^{4} + 3\cdot 41^{5} + 23\cdot 41^{6} + 23\cdot 41^{7} + 33\cdot 41^{8} + 33\cdot 41^{9} +O(41^{10})\) |
$r_{ 5 }$ | $=$ | \( 22 + 32\cdot 41 + 40\cdot 41^{2} + 31\cdot 41^{3} + 9\cdot 41^{4} + 37\cdot 41^{5} + 17\cdot 41^{6} + 17\cdot 41^{7} + 7\cdot 41^{8} + 7\cdot 41^{9} +O(41^{10})\) |
$r_{ 6 }$ | $=$ | \( 34 + 33\cdot 41 + 33\cdot 41^{2} + 14\cdot 41^{3} + 3\cdot 41^{4} + 41^{5} + 23\cdot 41^{6} + 11\cdot 41^{7} + 2\cdot 41^{8} + 23\cdot 41^{9} +O(41^{10})\) |
$r_{ 7 }$ | $=$ | \( 35 + 26\cdot 41 + 28\cdot 41^{2} + 24\cdot 41^{3} + 7\cdot 41^{4} + 28\cdot 41^{5} + 2\cdot 41^{6} + 6\cdot 41^{7} + 21\cdot 41^{8} + 23\cdot 41^{9} +O(41^{10})\) |
$r_{ 8 }$ | $=$ | \( 37 + 40\cdot 41^{2} + 3\cdot 41^{3} + 31\cdot 41^{4} + 32\cdot 41^{5} + 36\cdot 41^{6} + 13\cdot 41^{7} + 16\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
$2$ | $2$ | $(2,7)(4,5)$ | $0$ |
$1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
$1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ |
$2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
$2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $0$ |
$2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $0$ |
$2$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
$2$ | $8$ | $(1,7,3,5,8,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.