Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(1027\)\(\medspace = 13 \cdot 79 \) |
Artin field: | Galois closure of 3.3.1054729.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{1027}(529,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 342x - 989 \) . |
The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 22\cdot 23 + 18\cdot 23^{2} + 5\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\) |
$r_{ 2 }$ | $=$ | \( 5 + 2\cdot 23 + 4\cdot 23^{2} + 23^{3} + 2\cdot 23^{4} +O(23^{5})\) |
$r_{ 3 }$ | $=$ | \( 19 + 21\cdot 23 + 22\cdot 23^{2} + 15\cdot 23^{3} + 23^{4} +O(23^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $3$ | $(1,2,3)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,3,2)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.