Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(119\)\(\medspace = 7 \cdot 17 \) |
Artin field: | Galois closure of 6.6.11796113.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | even |
Dirichlet character: | \(\chi_{119}(67,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} - 17x^{4} + 11x^{3} + 57x^{2} - 25x - 13 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( a + 5 + \left(2 a + 7\right)\cdot 29 + \left(17 a + 3\right)\cdot 29^{2} + \left(8 a + 1\right)\cdot 29^{3} + \left(7 a + 17\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( a + 1 + \left(2 a + 24\right)\cdot 29 + 17 a\cdot 29^{2} + \left(8 a + 17\right)\cdot 29^{3} + \left(7 a + 16\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( a + 16 + \left(2 a + 27\right)\cdot 29 + \left(17 a + 15\right)\cdot 29^{2} + \left(8 a + 15\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 28 a + 10 + \left(26 a + 16\right)\cdot 29 + \left(11 a + 28\right)\cdot 29^{2} + \left(20 a + 26\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 28 a + 6 + \left(26 a + 4\right)\cdot 29 + \left(11 a + 26\right)\cdot 29^{2} + \left(20 a + 13\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 28 a + 21 + \left(26 a + 7\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(20 a + 12\right)\cdot 29^{3} + \left(21 a + 10\right)\cdot 29^{4} +O(29^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
$1$ | $3$ | $(1,2,3)(4,5,6)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,3,2)(4,6,5)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,5,3,4,2,6)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,6,2,4,3,5)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.