Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(10039\) |
Artin field: | Galois closure of 3.3.100781521.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{10039}(6307,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 3346x + 56144 \) . |
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 7.
Roots:
$r_{ 1 }$ | $=$ |
\( 11^{2} + 11^{3} + 6\cdot 11^{4} + 11^{5} + 5\cdot 11^{6} +O(11^{7})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 6\cdot 11 + 11^{3} + 6\cdot 11^{4} + 9\cdot 11^{5} + 4\cdot 11^{6} +O(11^{7})\)
| $r_{ 3 }$ |
$=$ |
\( 10 + 4\cdot 11 + 9\cdot 11^{2} + 8\cdot 11^{3} + 9\cdot 11^{4} + 10\cdot 11^{5} +O(11^{7})\)
| |
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.