Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(429\)\(\medspace = 3 \cdot 11 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.13343156541.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{429}(296,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 109x^{4} + 108x^{3} + 1878x^{2} - 107x - 5825 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 34 a + 28 + \left(30 a + 36\right)\cdot 53 + \left(11 a + 1\right)\cdot 53^{2} + \left(45 a + 41\right)\cdot 53^{3} + \left(33 a + 36\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 42 + \left(19 a + 39\right)\cdot 53 + \left(29 a + 5\right)\cdot 53^{2} + \left(31 a + 30\right)\cdot 53^{3} + \left(2 a + 50\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 19 a + 5 + \left(22 a + 19\right)\cdot 53 + \left(41 a + 17\right)\cdot 53^{2} + \left(7 a + 51\right)\cdot 53^{3} + \left(19 a + 20\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 a + 43 + \left(a + 11\right)\cdot 53 + \left(5 a + 32\right)\cdot 53^{2} + \left(42 a + 5\right)\cdot 53^{3} + \left(32 a + 19\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 17 + \left(51 a + 50\right)\cdot 53 + \left(47 a + 50\right)\cdot 53^{2} + \left(10 a + 9\right)\cdot 53^{3} + \left(20 a + 2\right)\cdot 53^{4} +O(53^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 44 a + 25 + \left(33 a + 1\right)\cdot 53 + \left(23 a + 51\right)\cdot 53^{2} + \left(21 a + 20\right)\cdot 53^{3} + \left(50 a + 29\right)\cdot 53^{4} +O(53^{5})\)
|
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 value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-1$ |
| $1$ | $3$ | $(1,2,4)(3,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,2)(3,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,6,4,3,2,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,2,3,4,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.