Basic invariants
Dimension: | $1$ |
Group: | $C_3$ |
Conductor: | \(10027\)\(\medspace = 37 \cdot 271 \) |
Artin field: | Galois closure of 3.3.100540729.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_3$ |
Parity: | even |
Dirichlet character: | \(\chi_{10027}(1654,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{3} - x^{2} - 3342x - 73160 \) . |
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 8.
Roots:
$r_{ 1 }$ | $=$ |
\( 4\cdot 5 + 2\cdot 5^{2} + 4\cdot 5^{3} + 2\cdot 5^{4} + 5^{5} + 3\cdot 5^{6} + 2\cdot 5^{7} +O(5^{8})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 3\cdot 5 + 4\cdot 5^{2} + 2\cdot 5^{4} + 4\cdot 5^{5} + 5^{6} + 4\cdot 5^{7} +O(5^{8})\)
| $r_{ 3 }$ |
$=$ |
\( 4 + 2\cdot 5 + 2\cdot 5^{2} + 4\cdot 5^{3} + 4\cdot 5^{4} + 3\cdot 5^{5} + 4\cdot 5^{6} + 2\cdot 5^{7} +O(5^{8})\)
| |
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.