Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(10003\)\(\medspace = 7 \cdot 1429 \) |
Artin field: | Galois closure of 3.3.100060009.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{10003}(4951,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 3334x + 39271 \) . |
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 7\cdot 23 + 11\cdot 23^{2} + 9\cdot 23^{3} + 6\cdot 23^{4} + 14\cdot 23^{5} +O(23^{6})\) |
$r_{ 2 }$ | $=$ | \( 9 + 10\cdot 23 + 12\cdot 23^{2} + 21\cdot 23^{3} + 8\cdot 23^{4} + 17\cdot 23^{5} +O(23^{6})\) |
$r_{ 3 }$ | $=$ | \( 10 + 5\cdot 23 + 22\cdot 23^{2} + 14\cdot 23^{3} + 7\cdot 23^{4} + 14\cdot 23^{5} +O(23^{6})\) |
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value |
$1$ | $1$ | $()$ | $1$ |
$1$ | $3$ | $(1,2,3)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.