Basic invariants
Dimension: | $1$ |
Group: | $C_4$ |
Conductor: | \(1003\)\(\medspace = 17 \cdot 59 \) |
Artin field: | Galois closure of 4.0.17102153.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_4$ |
Parity: | odd |
Dirichlet character: | \(\chi_{1003}(412,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 249x^{2} + x + 15046 \) . |
The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 18 + 49\cdot 83 + 18\cdot 83^{2} + 81\cdot 83^{3} + 54\cdot 83^{4} +O(83^{5})\) |
$r_{ 2 }$ | $=$ | \( 19 + 19\cdot 83 + 52\cdot 83^{2} + 3\cdot 83^{3} + 37\cdot 83^{4} +O(83^{5})\) |
$r_{ 3 }$ | $=$ | \( 56 + 52\cdot 83 + 29\cdot 83^{2} + 48\cdot 83^{3} + 62\cdot 83^{4} +O(83^{5})\) |
$r_{ 4 }$ | $=$ | \( 74 + 44\cdot 83 + 65\cdot 83^{2} + 32\cdot 83^{3} + 11\cdot 83^{4} +O(83^{5})\) |
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.