Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(4033\)\(\medspace = 37 \cdot 109 \) |
Artin field: | Galois closure of 3.3.16265089.2 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{4033}(935,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 1344x + 6871 \) . |
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 16 + 37\cdot 43 + 29\cdot 43^{2} + 40\cdot 43^{3} + 20\cdot 43^{4} +O(43^{5})\) |
$r_{ 2 }$ | $=$ | \( 29 + 9\cdot 43 + 38\cdot 43^{2} + 16\cdot 43^{3} + 39\cdot 43^{4} +O(43^{5})\) |
$r_{ 3 }$ | $=$ | \( 42 + 38\cdot 43 + 17\cdot 43^{2} + 28\cdot 43^{3} + 25\cdot 43^{4} +O(43^{5})\) |
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.