Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(91\)\(\medspace = 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.127353499.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{91}(69,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 21x^{4} - 22x^{3} + 58x^{2} + 23x + 155 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{2} + 102x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 39 a + 101 + \left(50 a + 31\right)\cdot 103 + \left(76 a + 27\right)\cdot 103^{2} + \left(51 a + 63\right)\cdot 103^{3} + \left(29 a + 69\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 64 a + 37 + \left(52 a + 43\right)\cdot 103 + \left(26 a + 53\right)\cdot 103^{2} + \left(51 a + 38\right)\cdot 103^{3} + \left(73 a + 47\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 98 a + 27 + \left(51 a + 56\right)\cdot 103 + \left(11 a + 87\right)\cdot 103^{2} + \left(28 a + 31\right)\cdot 103^{3} + \left(59 a + 101\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a + 100 + \left(53 a + 69\right)\cdot 103 + \left(99 a + 23\right)\cdot 103^{2} + \left(66 a + 28\right)\cdot 103^{3} + \left(44 a + 93\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 22 + \left(51 a + 10\right)\cdot 103 + \left(91 a + 47\right)\cdot 103^{2} + \left(74 a + 48\right)\cdot 103^{3} + \left(43 a + 29\right)\cdot 103^{4} +O(103^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 77 a + 23 + \left(49 a + 97\right)\cdot 103 + \left(3 a + 69\right)\cdot 103^{2} + \left(36 a + 98\right)\cdot 103^{3} + \left(58 a + 70\right)\cdot 103^{4} +O(103^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,4,5)(2,6,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,5,4)(2,3,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,4,2,5,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,2,4,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.