Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(143\)\(\medspace = 11 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.38014691.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{143}(120,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 5x^{3} + 39x^{2} - 74x + 155 \)
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The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 72 a + 14 + \left(59 a + 12\right)\cdot 73 + \left(31 a + 12\right)\cdot 73^{2} + \left(37 a + 21\right)\cdot 73^{3} + \left(62 a + 30\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 72 a + 9 + \left(59 a + 65\right)\cdot 73 + \left(31 a + 48\right)\cdot 73^{2} + \left(37 a + 53\right)\cdot 73^{3} + \left(62 a + 43\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 11 + \left(13 a + 47\right)\cdot 73 + \left(41 a + 47\right)\cdot 73^{2} + \left(35 a + 28\right)\cdot 73^{3} + \left(10 a + 34\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( a + 52 + \left(13 a + 14\right)\cdot 73 + \left(41 a + 67\right)\cdot 73^{2} + \left(35 a + 30\right)\cdot 73^{3} + \left(10 a + 33\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 6 + \left(13 a + 27\right)\cdot 73 + \left(41 a + 11\right)\cdot 73^{2} + \left(35 a + 61\right)\cdot 73^{3} + \left(10 a + 47\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 72 a + 55 + \left(59 a + 52\right)\cdot 73 + \left(31 a + 31\right)\cdot 73^{2} + \left(37 a + 23\right)\cdot 73^{3} + \left(62 a + 29\right)\cdot 73^{4} +O(73^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,2,3,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,3,2,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.