Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(133\)\(\medspace = 7 \cdot 19 \) |
Artin field: | Galois closure of 6.0.16468459.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{133}(37,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 10x^{4} - 7x^{3} + 75x^{2} - 16x + 239 \) . |
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:
$r_{ 1 }$ | $=$ |
\( 8 a + \left(a + 1\right)\cdot 13 + \left(7 a + 5\right)\cdot 13^{2} + \left(6 a + 1\right)\cdot 13^{3} + \left(11 a + 12\right)\cdot 13^{4} +O(13^{5})\)
$r_{ 2 }$ |
$=$ |
\( 8 a + 10 + \left(a + 2\right)\cdot 13 + \left(7 a + 6\right)\cdot 13^{2} + \left(6 a + 7\right)\cdot 13^{3} + \left(11 a + 8\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 5 a + 8 + \left(11 a + 7\right)\cdot 13 + \left(5 a + 10\right)\cdot 13^{2} + 6 a\cdot 13^{3} + \left(a + 4\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 5 a + 5 + \left(11 a + 9\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + \left(6 a + 6\right)\cdot 13^{3} + a\cdot 13^{4} +O(13^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 8 a + 11 + \left(a + 5\right)\cdot 13 + \left(7 a + 6\right)\cdot 13^{2} + \left(6 a + 11\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} +O(13^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 5 a + 6 + \left(11 a + 12\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + \left(6 a + 10\right)\cdot 13^{3} + \left(a + 2\right)\cdot 13^{4} +O(13^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
$1$ | $3$ | $(1,2,5)(3,4,6)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,5,2)(3,6,4)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,4,5,3,2,6)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,6,2,3,5,4)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.