Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(155\)\(\medspace = 5 \cdot 31 \) |
Artin field: | Galois closure of 6.0.3578643875.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{155}(119,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{6} - x^{5} + 34x^{4} - 73x^{3} + 509x^{2} - 470x + 1396 \) . |
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:
$r_{ 1 }$ | $=$ | \( 5 a + 11 + \left(14 a + 13\right)\cdot 29 + \left(26 a + 16\right)\cdot 29^{2} + \left(15 a + 19\right)\cdot 29^{3} + \left(19 a + 5\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 7 a + 14 + \left(a + 10\right)\cdot 29 + \left(26 a + 10\right)\cdot 29^{2} + \left(9 a + 27\right)\cdot 29^{3} + \left(28 a + 2\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 26 a + 11 + 12\cdot 29 + \left(15 a + 15\right)\cdot 29^{2} + \left(a + 5\right)\cdot 29^{3} + \left(8 a + 10\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 3 a + 25 + \left(28 a + 19\right)\cdot 29 + \left(13 a + 2\right)\cdot 29^{2} + \left(27 a + 27\right)\cdot 29^{3} + \left(20 a + 19\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 24 a + 7 + \left(14 a + 21\right)\cdot 29 + \left(2 a + 18\right)\cdot 29^{2} + \left(13 a + 14\right)\cdot 29^{3} + 9 a\cdot 29^{4} +O(29^{5})\) |
$r_{ 6 }$ | $=$ | \( 22 a + 20 + \left(27 a + 9\right)\cdot 29 + \left(2 a + 23\right)\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
$1$ | $3$ | $(1,3,6)(2,5,4)$ | $-\zeta_{3} - 1$ |
$1$ | $3$ | $(1,6,3)(2,4,5)$ | $\zeta_{3}$ |
$1$ | $6$ | $(1,2,3,5,6,4)$ | $-\zeta_{3}$ |
$1$ | $6$ | $(1,4,6,5,3,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.