Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
Artin number field: | Galois closure of 6.6.996974433.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | even |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ |
\( 9 a + 3 + \left(23 a + 20\right)\cdot 29 + \left(18 a + 14\right)\cdot 29^{2} + 22\cdot 29^{3} + \left(17 a + 14\right)\cdot 29^{4} +O(29^{5})\)
$r_{ 2 }$ |
$=$ |
\( 5 a + 22 + \left(20 a + 15\right)\cdot 29 + \left(2 a + 3\right)\cdot 29^{2} + \left(22 a + 26\right)\cdot 29^{3} + \left(6 a + 24\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 3 a + 20 + \left(13 a + 19\right)\cdot 29 + \left(9 a + 19\right)\cdot 29^{2} + 10\cdot 29^{3} + \left(3 a + 6\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 26 a + 6 + \left(15 a + 24\right)\cdot 29 + \left(19 a + 24\right)\cdot 29^{2} + \left(28 a + 2\right)\cdot 29^{3} + \left(25 a + 21\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 24 a + 18 + \left(8 a + 24\right)\cdot 29 + \left(26 a + 25\right)\cdot 29^{2} + \left(6 a + 17\right)\cdot 29^{3} + \left(22 a + 7\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 20 a + 19 + \left(5 a + 11\right)\cdot 29 + \left(10 a + 27\right)\cdot 29^{2} + \left(28 a + 6\right)\cdot 29^{3} + \left(11 a + 12\right)\cdot 29^{4} +O(29^{5})\)
| |
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 values | |
$c1$ | $c2$ | |||
$1$ | $1$ | $()$ | $1$ | $1$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ | $-1$ |
$1$ | $3$ | $(1,3,2)(4,5,6)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,2,3)(4,6,5)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,5,3,6,2,4)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,4,2,6,3,5)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |