Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Artin number field: | Galois closure of 6.0.562166163.1 |
| 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:
| $r_{ 1 }$ | $=$ |
\( 12 a + 2 + \left(10 a + 17\right)\cdot 23 + \left(15 a + 7\right)\cdot 23^{2} + \left(10 a + 4\right)\cdot 23^{3} + \left(16 a + 1\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 a + 12 + \left(20 a + 22\right)\cdot 23 + \left(4 a + 1\right)\cdot 23^{2} + \left(a + 12\right)\cdot 23^{3} + 11\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 4 + \left(2 a + 22\right)\cdot 23 + \left(18 a + 13\right)\cdot 23^{2} + \left(21 a + 9\right)\cdot 23^{3} + \left(22 a + 10\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 3 + \left(12 a + 3\right)\cdot 23 + \left(7 a + 5\right)\cdot 23^{2} + \left(12 a + 10\right)\cdot 23^{3} + 6 a\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 a + 17 + \left(10 a + 6\right)\cdot 23 + \left(12 a + 1\right)\cdot 23^{2} + \left(13 a + 9\right)\cdot 23^{3} + \left(6 a + 11\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a + 8 + \left(12 a + 20\right)\cdot 23 + \left(10 a + 15\right)\cdot 23^{2} + 9 a\cdot 23^{3} + \left(16 a + 11\right)\cdot 23^{4} +O(23^{5})\)
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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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,5,4,3,6)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,3,4,5,2)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |