Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(161\)\(\medspace = 7 \cdot 23 \) |
| Artin field: | Galois closure of 6.6.204490769.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{161}(45,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} - 41x^{4} + 41x^{3} + 463x^{2} - 463x - 1049 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a + 8 + 12\cdot 13 + \left(7 a + 4\right)\cdot 13^{2} + 7\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 5 + 8 a\cdot 13 + \left(a + 12\right)\cdot 13^{2} + 6\cdot 13^{3} + \left(4 a + 5\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 5 a\cdot 13 + \left(9 a + 7\right)\cdot 13^{2} + \left(a + 6\right)\cdot 13^{3} + \left(8 a + 2\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 1 + 4 a\cdot 13 + \left(11 a + 5\right)\cdot 13^{2} + \left(12 a + 5\right)\cdot 13^{3} + \left(8 a + 9\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a + 3 + \left(7 a + 2\right)\cdot 13 + \left(3 a + 11\right)\cdot 13^{2} + \left(11 a + 11\right)\cdot 13^{3} + \left(4 a + 8\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a + 10 + \left(12 a + 10\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + 12 a\cdot 13^{3} + 11 a\cdot 13^{4} +O(13^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ | ✓ |
| $1$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ | |
| $1$ | $3$ | $(1,3,4)(2,6,5)$ | $\zeta_{3}$ | |
| $1$ | $3$ | $(1,4,3)(2,5,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $6$ | $(1,2,3,6,4,5)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,5,4,6,3,2)$ | $-\zeta_{3}$ |