Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(532\)\(\medspace = 2^{2} \cdot 7 \cdot 19 \) |
| Artin number field: | Galois closure of 6.0.2663410937152.1 |
| 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 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 20 + \left(9 a + 3\right)\cdot 23 + \left(a + 3\right)\cdot 23^{2} + \left(22 a + 13\right)\cdot 23^{3} + \left(17 a + 4\right)\cdot 23^{4} + \left(6 a + 2\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a + 8 + \left(7 a + 11\right)\cdot 23 + \left(a + 2\right)\cdot 23^{2} + \left(2 a + 10\right)\cdot 23^{3} + \left(21 a + 14\right)\cdot 23^{4} + \left(2 a + 7\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 9 + \left(20 a + 9\right)\cdot 23 + \left(2 a + 7\right)\cdot 23^{2} + \left(14 a + 10\right)\cdot 23^{3} + \left(10 a + 19\right)\cdot 23^{4} + \left(20 a + 7\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 3 + \left(13 a + 19\right)\cdot 23 + \left(21 a + 19\right)\cdot 23^{2} + 9\cdot 23^{3} + \left(5 a + 18\right)\cdot 23^{4} + \left(16 a + 20\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a + 15 + \left(15 a + 11\right)\cdot 23 + \left(21 a + 20\right)\cdot 23^{2} + \left(20 a + 12\right)\cdot 23^{3} + \left(a + 8\right)\cdot 23^{4} + \left(20 a + 15\right)\cdot 23^{5} +O(23^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 14 + \left(2 a + 13\right)\cdot 23 + \left(20 a + 15\right)\cdot 23^{2} + \left(8 a + 12\right)\cdot 23^{3} + \left(12 a + 3\right)\cdot 23^{4} + \left(2 a + 15\right)\cdot 23^{5} +O(23^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |