Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(124\)\(\medspace = 2^{2} \cdot 31 \) |
| Artin field: | Galois closure of 6.0.59105344.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{124}(67,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 21x^{4} + 116x^{2} + 64 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 20 + \left(a + 11\right)\cdot 23 + \left(22 a + 1\right)\cdot 23^{2} + \left(12 a + 21\right)\cdot 23^{3} + \left(20 a + 8\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 17 + \left(7 a + 18\right)\cdot 23 + \left(7 a + 7\right)\cdot 23^{2} + \left(21 a + 5\right)\cdot 23^{3} + \left(14 a + 7\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 9 + \left(7 a + 22\right)\cdot 23 + \left(8 a + 6\right)\cdot 23^{2} + \left(17 a + 21\right)\cdot 23^{3} + \left(a + 6\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 3 + \left(21 a + 11\right)\cdot 23 + 21\cdot 23^{2} + \left(10 a + 1\right)\cdot 23^{3} + \left(2 a + 14\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 17 a + 6 + \left(15 a + 4\right)\cdot 23 + \left(15 a + 15\right)\cdot 23^{2} + \left(a + 17\right)\cdot 23^{3} + \left(8 a + 15\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 14 + 15 a\cdot 23 + \left(14 a + 16\right)\cdot 23^{2} + \left(5 a + 1\right)\cdot 23^{3} + \left(21 a + 16\right)\cdot 23^{4} +O(23^{5})\)
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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 value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.