Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Artin field: | Galois closure of 6.6.2436053373.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{117}(4,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 39x^{4} - 26x^{3} + 351x^{2} + 234x - 468 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{2} + 58x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a + 40 + \left(24 a + 2\right)\cdot 59 + \left(22 a + 30\right)\cdot 59^{2} + \left(18 a + 54\right)\cdot 59^{3} + \left(17 a + 18\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 a + 9 + \left(26 a + 55\right)\cdot 59 + \left(58 a + 57\right)\cdot 59^{2} + \left(21 a + 43\right)\cdot 59^{3} + \left(33 a + 15\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 10 + \left(8 a + 1\right)\cdot 59 + \left(37 a + 30\right)\cdot 59^{2} + \left(18 a + 19\right)\cdot 59^{3} + \left(8 a + 24\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 a + 21 + \left(50 a + 57\right)\cdot 59 + \left(21 a + 58\right)\cdot 59^{2} + 40 a\cdot 59^{3} + \left(50 a + 14\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 56 a + 43 + \left(34 a + 23\right)\cdot 59 + \left(36 a + 28\right)\cdot 59^{2} + \left(40 a + 50\right)\cdot 59^{3} + \left(41 a + 17\right)\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 a + 54 + \left(32 a + 36\right)\cdot 59 + 30\cdot 59^{2} + \left(37 a + 7\right)\cdot 59^{3} + \left(25 a + 27\right)\cdot 59^{4} +O(59^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ | |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,6,3,5,2,4)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,4,2,5,3,6)$ | $-\zeta_{3}$ |