Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(1009\) |
Artin field: | Galois closure of 3.3.1018081.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{1009}(634,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 336x + 1719 \) . |
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 5\cdot 13 + 12\cdot 13^{2} + 13^{3} + 4\cdot 13^{4} + 12\cdot 13^{5} +O(13^{6})\) |
$r_{ 2 }$ | $=$ | \( 10 + 2\cdot 13 + 3\cdot 13^{2} + 3\cdot 13^{4} + 6\cdot 13^{5} +O(13^{6})\) |
$r_{ 3 }$ | $=$ | \( 11 + 4\cdot 13 + 10\cdot 13^{2} + 10\cdot 13^{3} + 5\cdot 13^{4} + 7\cdot 13^{5} +O(13^{6})\) |
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.