Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \) |
| Artin field: | Galois closure of 6.0.19208000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{140}(39,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + 12x^{4} - 14x^{3} + 87x^{2} - 64x + 281 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( a + 11 + \left(12 a + 4\right)\cdot 13 + \left(11 a + 10\right)\cdot 13^{2} + \left(12 a + 7\right)\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a + 12 + 2\cdot 13 + \left(a + 10\right)\cdot 13^{2} + 8\cdot 13^{3} + 12\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( a + 9 + \left(12 a + 9\right)\cdot 13 + \left(11 a + 11\right)\cdot 13^{2} + \left(12 a + 4\right)\cdot 13^{3} + \left(12 a + 11\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a + 6\cdot 13 + \left(a + 10\right)\cdot 13^{2} + 12\cdot 13^{3} + 13^{4} +O(13^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( a + 12 + \left(12 a + 7\right)\cdot 13 + \left(11 a + 10\right)\cdot 13^{2} + \left(12 a + 11\right)\cdot 13^{3} + \left(12 a + 1\right)\cdot 13^{4} +O(13^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a + 10 + 7\cdot 13 + \left(a + 11\right)\cdot 13^{2} + 5\cdot 13^{3} + 11\cdot 13^{4} +O(13^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $1$ | |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ | ✓ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $-\zeta_{3} - 1$ | |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $\zeta_{3}$ | |
| $1$ | $6$ | $(1,4,3,2,5,6)$ | $\zeta_{3} + 1$ | |
| $1$ | $6$ | $(1,6,5,2,3,4)$ | $-\zeta_{3}$ |