Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(133\)\(\medspace = 7 \cdot 19 \) |
| Artin number field: | Galois closure of 6.0.44700103.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a + 23 + \left(25 a + 20\right)\cdot 31 + \left(17 a + 13\right)\cdot 31^{2} + \left(8 a + 15\right)\cdot 31^{3} + \left(27 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a + 15 + \left(25 a + 12\right)\cdot 31 + \left(17 a + 13\right)\cdot 31^{2} + \left(8 a + 17\right)\cdot 31^{3} + \left(27 a + 21\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a + 24 + \left(5 a + 12\right)\cdot 31 + \left(13 a + 23\right)\cdot 31^{2} + \left(22 a + 16\right)\cdot 31^{3} + \left(3 a + 5\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 a + 20 + \left(5 a + 28\right)\cdot 31 + \left(13 a + 29\right)\cdot 31^{2} + \left(22 a + 13\right)\cdot 31^{3} + \left(3 a + 29\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a + 1 + \left(5 a + 21\right)\cdot 31 + \left(13 a + 23\right)\cdot 31^{2} + \left(22 a + 14\right)\cdot 31^{3} + \left(3 a + 18\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 20 a + 11 + \left(25 a + 28\right)\cdot 31 + \left(17 a + 19\right)\cdot 31^{2} + \left(8 a + 14\right)\cdot 31^{3} + \left(27 a + 14\right)\cdot 31^{4} +O(31^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,2,5,6,3)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,5,2,4)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |