Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(315\)\(\medspace = 3^{2} \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.6.1969120125.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{315}(79,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 42x^{4} + 19x^{3} + 483x^{2} + 732x + 251 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 21 a + 16 + \left(9 a + 15\right)\cdot 23 + \left(8 a + 4\right)\cdot 23^{2} + \left(2 a + 8\right)\cdot 23^{3} + \left(18 a + 9\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 5 + \left(9 a + 17\right)\cdot 23 + \left(8 a + 19\right)\cdot 23^{2} + \left(2 a + 18\right)\cdot 23^{3} + \left(18 a + 11\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 1 + \left(13 a + 16\right)\cdot 23 + \left(14 a + 3\right)\cdot 23^{2} + \left(20 a + 15\right)\cdot 23^{3} + \left(4 a + 22\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 2 a + 17 + \left(13 a + 13\right)\cdot 23 + \left(14 a + 6\right)\cdot 23^{2} + \left(20 a + 9\right)\cdot 23^{3} + \left(4 a + 19\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 12 + \left(13 a + 14\right)\cdot 23 + \left(14 a + 11\right)\cdot 23^{2} + \left(20 a + 4\right)\cdot 23^{3} + \left(4 a + 20\right)\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a + 21 + \left(9 a + 14\right)\cdot 23 + \left(8 a + 22\right)\cdot 23^{2} + \left(2 a + 12\right)\cdot 23^{3} + \left(18 a + 8\right)\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.