Basic invariants
| Dimension: | $1$ |
| Group: | $C_{10}$ |
| Conductor: | \(88\)\(\medspace = 2^{3} \cdot 11 \) |
| Artin number field: | Galois closure of 10.0.7024111812608.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_{10}$ |
| 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 $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{5} + x + 14 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( a^{4} + 14 a^{3} + 15 a^{2} + 14 a + 13 + \left(12 a^{4} + 5 a^{3} + 10 a^{2} + 4 a + 12\right)\cdot 17 + \left(8 a^{4} + 16 a^{3} + 6 a^{2} + 2 a + 14\right)\cdot 17^{2} + \left(13 a^{4} + 14 a^{3} + 7 a^{2} + 12 a + 3\right)\cdot 17^{3} + \left(9 a^{4} + 3 a^{3} + 13 a^{2} + 8 a + 16\right)\cdot 17^{4} + \left(13 a^{4} + 7 a^{3} + 16 a^{2} + 13 a + 3\right)\cdot 17^{5} + \left(16 a^{4} + 11 a^{3} + 13 a^{2} + 2 a + 10\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 a^{4} + 13 a^{3} + 7 a^{2} + 5 a + 14 + \left(9 a^{4} + 11 a^{3} + 9 a\right)\cdot 17 + \left(10 a^{4} + 3 a^{3} + 3 a^{2} + 9 a + 4\right)\cdot 17^{2} + \left(12 a^{4} + 12 a^{3} + 8 a^{2} + 14 a + 10\right)\cdot 17^{3} + \left(15 a^{4} + 4 a^{3} + 7 a^{2} + 8 a + 7\right)\cdot 17^{4} + \left(10 a^{4} + 12 a^{3} + 12 a^{2} + a + 5\right)\cdot 17^{5} + \left(4 a^{4} + 11 a^{3} + 8 a^{2} + 16 a + 10\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a^{4} + 14 a^{3} + 8 a^{2} + 9 a + 1 + \left(9 a^{4} + 2 a^{3} + 5 a^{2} + 12 a + 7\right)\cdot 17 + \left(7 a^{4} + 12 a^{3} + 4 a^{2} + 14 a + 10\right)\cdot 17^{2} + \left(8 a^{4} + 10 a^{3} + 13 a^{2} + 14 a + 6\right)\cdot 17^{3} + \left(13 a^{4} + 16 a^{3} + 14 a + 12\right)\cdot 17^{4} + \left(4 a^{4} + 14 a^{3} + 7 a + 3\right)\cdot 17^{5} + \left(5 a^{3} + 2 a^{2} + 14 a + 7\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a^{4} + 8 a^{3} + 5 a^{2} + 16 a + 12 + \left(7 a^{4} + 11 a^{3} + 2 a^{2} + 14 a + 15\right)\cdot 17 + \left(4 a^{4} + 14 a^{3} + 13 a^{2} + a + 7\right)\cdot 17^{2} + \left(2 a^{4} + 7 a^{3} + a^{2} + 4 a + 8\right)\cdot 17^{3} + \left(5 a^{4} + 7 a^{3} + 3 a^{2} + 8 a + 12\right)\cdot 17^{4} + \left(5 a^{4} + 12 a^{3} + 13 a\right)\cdot 17^{5} + \left(9 a^{4} + 5 a^{3} + 2 a^{2} + 6 a + 11\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a^{4} + 6 a^{3} + 4 a^{2} + 2 + \left(2 a^{4} + 8 a^{3} + a^{2} + 6 a + 2\right)\cdot 17 + \left(10 a^{4} + 5 a^{3} + 4 a^{2} + 9 a + 7\right)\cdot 17^{2} + \left(11 a^{4} + 4 a^{3} + 14 a^{2} + 16 a + 9\right)\cdot 17^{3} + \left(12 a^{4} + 6 a^{3} + 9 a^{2} + 5 a + 8\right)\cdot 17^{4} + \left(12 a^{4} + 5 a^{3} + 4 a^{2} + 16 a + 3\right)\cdot 17^{5} + \left(4 a^{4} + 11 a^{3} + 9 a^{2} + 7 a\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a^{4} + 11 a^{3} + 13 a^{2} + 15 + \left(14 a^{4} + 8 a^{3} + 15 a^{2} + 11 a + 14\right)\cdot 17 + \left(6 a^{4} + 11 a^{3} + 12 a^{2} + 7 a + 9\right)\cdot 17^{2} + \left(5 a^{4} + 12 a^{3} + 2 a^{2} + 7\right)\cdot 17^{3} + \left(4 a^{4} + 10 a^{3} + 7 a^{2} + 11 a + 8\right)\cdot 17^{4} + \left(4 a^{4} + 11 a^{3} + 12 a^{2} + 13\right)\cdot 17^{5} + \left(12 a^{4} + 5 a^{3} + 7 a^{2} + 9 a + 16\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a^{4} + 9 a^{3} + 12 a^{2} + a + 5 + \left(9 a^{4} + 5 a^{3} + 14 a^{2} + 2 a + 1\right)\cdot 17 + \left(12 a^{4} + 2 a^{3} + 3 a^{2} + 15 a + 9\right)\cdot 17^{2} + \left(14 a^{4} + 9 a^{3} + 15 a^{2} + 12 a + 8\right)\cdot 17^{3} + \left(11 a^{4} + 9 a^{3} + 13 a^{2} + 8 a + 4\right)\cdot 17^{4} + \left(11 a^{4} + 4 a^{3} + 16 a^{2} + 3 a + 16\right)\cdot 17^{5} + \left(7 a^{4} + 11 a^{3} + 14 a^{2} + 10 a + 5\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 14 a^{4} + 3 a^{3} + 9 a^{2} + 8 a + 16 + \left(7 a^{4} + 14 a^{3} + 11 a^{2} + 4 a + 9\right)\cdot 17 + \left(9 a^{4} + 4 a^{3} + 12 a^{2} + 2 a + 6\right)\cdot 17^{2} + \left(8 a^{4} + 6 a^{3} + 3 a^{2} + 2 a + 10\right)\cdot 17^{3} + \left(3 a^{4} + 16 a^{2} + 2 a + 4\right)\cdot 17^{4} + \left(12 a^{4} + 2 a^{3} + 16 a^{2} + 9 a + 13\right)\cdot 17^{5} + \left(16 a^{4} + 11 a^{3} + 14 a^{2} + 2 a + 9\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 9 }$ | $=$ |
\( 14 a^{4} + 4 a^{3} + 10 a^{2} + 12 a + 3 + \left(7 a^{4} + 5 a^{3} + 16 a^{2} + 7 a + 16\right)\cdot 17 + \left(6 a^{4} + 13 a^{3} + 13 a^{2} + 7 a + 12\right)\cdot 17^{2} + \left(4 a^{4} + 4 a^{3} + 8 a^{2} + 2 a + 6\right)\cdot 17^{3} + \left(a^{4} + 12 a^{3} + 9 a^{2} + 8 a + 9\right)\cdot 17^{4} + \left(6 a^{4} + 4 a^{3} + 4 a^{2} + 15 a + 11\right)\cdot 17^{5} + \left(12 a^{4} + 5 a^{3} + 8 a^{2} + 6\right)\cdot 17^{6} +O(17^{7})\)
|
| $r_{ 10 }$ | $=$ |
\( 16 a^{4} + 3 a^{3} + 2 a^{2} + 3 a + 4 + \left(4 a^{4} + 11 a^{3} + 6 a^{2} + 12 a + 4\right)\cdot 17 + \left(8 a^{4} + 10 a^{2} + 14 a + 2\right)\cdot 17^{2} + \left(3 a^{4} + 2 a^{3} + 9 a^{2} + 4 a + 13\right)\cdot 17^{3} + \left(7 a^{4} + 13 a^{3} + 3 a^{2} + 8 a\right)\cdot 17^{4} + \left(3 a^{4} + 9 a^{3} + 3 a + 13\right)\cdot 17^{5} + \left(5 a^{3} + 3 a^{2} + 14 a + 6\right)\cdot 17^{6} +O(17^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,10)(2,9)(3,8)(4,7)(5,6)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $5$ | $(1,4,9,3,6)(2,8,5,10,7)$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,9,6,4,3)(2,5,7,8,10)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,3,4,6,9)(2,10,8,7,5)$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,6,3,9,4)(2,7,10,5,8)$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ |
| $1$ | $10$ | $(1,7,9,8,6,10,4,2,3,5)$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
| $1$ | $10$ | $(1,8,4,5,9,10,3,7,6,2)$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ |
| $1$ | $10$ | $(1,2,6,7,3,10,9,5,4,8)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ |
| $1$ | $10$ | $(1,5,3,2,4,10,6,8,9,7)$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ |