Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(323\)\(\medspace = 17 \cdot 19 \) |
Artin field: | Galois closure of 4.0.1773593.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{323}(132,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 79x^{2} + x + 1616 \) . |
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 9\cdot 47 + 42\cdot 47^{2} + 36\cdot 47^{3} + 9\cdot 47^{4} +O(47^{5})\) |
$r_{ 2 }$ | $=$ | \( 22 + 36\cdot 47 + 39\cdot 47^{2} + 40\cdot 47^{3} + 24\cdot 47^{4} +O(47^{5})\) |
$r_{ 3 }$ | $=$ | \( 24 + 24\cdot 47 + 45\cdot 47^{2} + 30\cdot 47^{3} + 14\cdot 47^{4} +O(47^{5})\) |
$r_{ 4 }$ | $=$ | \( 43 + 23\cdot 47 + 13\cdot 47^{2} + 32\cdot 47^{3} + 44\cdot 47^{4} +O(47^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
$1$ | $4$ | $(1,3,2,4)$ | $\zeta_{4}$ |
$1$ | $4$ | $(1,4,2,3)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.