Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(117\)\(\medspace = 3^{2} \cdot 13 \) |
| Artin field: | Galois closure of 6.6.14414517.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{117}(103,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} - 12x^{4} + 27x^{3} + 21x^{2} - 48x + 17 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 12 + \left(14 a + 1\right)\cdot 19 + \left(6 a + 12\right)\cdot 19^{2} + \left(a + 1\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a + 14 + \left(14 a + 14\right)\cdot 19 + \left(6 a + 9\right)\cdot 19^{2} + 14\cdot 19^{3} + \left(a + 15\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 11 + \left(4 a + 13\right)\cdot 19 + \left(12 a + 1\right)\cdot 19^{2} + \left(18 a + 8\right)\cdot 19^{3} + \left(17 a + 16\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a + 9 + 4 a\cdot 19 + \left(12 a + 4\right)\cdot 19^{2} + \left(18 a + 13\right)\cdot 19^{3} + \left(17 a + 1\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 18 + \left(14 a + 13\right)\cdot 19 + \left(6 a + 18\right)\cdot 19^{2} + 3\cdot 19^{3} + \left(a + 1\right)\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a + 15 + \left(4 a + 12\right)\cdot 19 + \left(12 a + 10\right)\cdot 19^{2} + \left(18 a + 16\right)\cdot 19^{3} + \left(17 a + 1\right)\cdot 19^{4} +O(19^{5})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ | |
| $1$ | $3$ | $(1,2,5)(3,6,4)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,5,2)(3,4,6)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,3,5,4,2,6)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,6,2,4,5,3)$ | $-\zeta_{3}$ |