Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(104\)\(\medspace = 2^{3} \cdot 13 \) |
Artin field: | Galois closure of 4.0.140608.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{104}(5,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + 26x^{2} + 52 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 2 + 24\cdot 43 + 9\cdot 43^{2} + 3\cdot 43^{3} + 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 20 + 21\cdot 43 + 13\cdot 43^{3} + 8\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 23 + 21\cdot 43 + 42\cdot 43^{2} + 29\cdot 43^{3} + 34\cdot 43^{4} +O(43^{5})\) |
$r_{ 4 }$ | $=$ | \( 41 + 18\cdot 43 + 33\cdot 43^{2} + 39\cdot 43^{3} + 41\cdot 43^{4} +O(43^{5})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
$1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.