Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Artin number field: | Galois closure of 6.0.7308160119.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 9 + \left(a + 14\right)\cdot 23 + \left(22 a + 4\right)\cdot 23^{2} + \left(a + 4\right)\cdot 23^{3} + \left(3 a + 21\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 12 + \left(9 a + 1\right)\cdot 23 + \left(3 a + 18\right)\cdot 23^{2} + \left(10 a + 18\right)\cdot 23^{3} + \left(14 a + 12\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 2 + \left(11 a + 7\right)\cdot 23 + 20 a\cdot 23^{2} + 10 a\cdot 23^{3} + \left(5 a + 12\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a + 5 + \left(11 a + 17\right)\cdot 23 + \left(2 a + 6\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(17 a + 12\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a + 6 + \left(13 a + 1\right)\cdot 23 + \left(19 a + 15\right)\cdot 23^{2} + \left(12 a + 12\right)\cdot 23^{3} + \left(8 a + 8\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a + 12 + \left(21 a + 4\right)\cdot 23 + 23^{2} + \left(21 a + 9\right)\cdot 23^{3} + \left(19 a + 2\right)\cdot 23^{4} +O(23^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,3,6,2,4)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,6,3,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |