Basic invariants
| Dimension: | $1$ |
| Group: | $C_3$ |
| Conductor: | \(31\) |
| Artin field: | Galois closure of 3.3.961.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_3$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{31}(5,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - x^{2} - 10x + 8 \)
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The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 4\cdot 23 + 5\cdot 23^{2} + 21\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
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| $r_{ 2 }$ | $=$ |
\( 17 + 2\cdot 23 + 10\cdot 23^{2} + 11\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 15\cdot 23 + 7\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})\)
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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.