Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(148\)\(\medspace = 2^{2} \cdot 37 \) |
Artin field: | Galois closure of 6.0.4438013248.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{148}(27,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} + 37x^{4} + 74x^{2} + 37 \) . |
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:
$r_{ 1 }$ | $=$ | \( 6 a + 14 + \left(13 a + 13\right)\cdot 29 + \left(26 a + 27\right)\cdot 29^{2} + \left(8 a + 19\right)\cdot 29^{3} + \left(9 a + 24\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 a + 26 + \left(19 a + 27\right)\cdot 29 + \left(7 a + 4\right)\cdot 29^{2} + \left(15 a + 9\right)\cdot 29^{3} + \left(27 a + 11\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 13 a + 11 + \left(12 a + 4\right)\cdot 29 + \left(12 a + 4\right)\cdot 29^{2} + \left(6 a + 19\right)\cdot 29^{3} + \left(18 a + 15\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 23 a + 15 + \left(15 a + 15\right)\cdot 29 + \left(2 a + 1\right)\cdot 29^{2} + \left(20 a + 9\right)\cdot 29^{3} + \left(19 a + 4\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 22 a + 3 + \left(9 a + 1\right)\cdot 29 + \left(21 a + 24\right)\cdot 29^{2} + \left(13 a + 19\right)\cdot 29^{3} + \left(a + 17\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 6 }$ | $=$ | \( 16 a + 18 + \left(16 a + 24\right)\cdot 29 + \left(16 a + 24\right)\cdot 29^{2} + \left(22 a + 9\right)\cdot 29^{3} + \left(10 a + 13\right)\cdot 29^{4} +O(29^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
$1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ |
$1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ |
$1$ | $6$ | $(1,3,5,4,6,2)$ | $\zeta_{3} + 1$ |
$1$ | $6$ | $(1,2,6,4,5,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.