Basic invariants
| Dimension: | $1$ |
| Group: | $C_{16}$ |
| Conductor: | \(64\)\(\medspace = 2^{6} \) |
| Artin field: | Galois closure of 16.0.604462909807314587353088.1 |
| Galois orbit size: | $8$ |
| Smallest permutation container: | $C_{16}$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{64}(51,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{16} + 16x^{14} + 104x^{12} + 352x^{10} + 660x^{8} + 672x^{6} + 336x^{4} + 64x^{2} + 2 \)
|
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 20\cdot 31 + 4\cdot 31^{2} + 4\cdot 31^{3} + 19\cdot 31^{4} + 21\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 12\cdot 31 + 19\cdot 31^{2} + 8\cdot 31^{3} + 12\cdot 31^{4} + 19\cdot 31^{5} + 29\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 + 9\cdot 31 + 31^{2} + 22\cdot 31^{3} + 25\cdot 31^{4} + 3\cdot 31^{5} + 9\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 + 2\cdot 31 + 27\cdot 31^{2} + 26\cdot 31^{3} + 18\cdot 31^{4} + 16\cdot 31^{5} + 16\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 + 4\cdot 31 + 11\cdot 31^{2} + 23\cdot 31^{4} + 17\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 + 30\cdot 31 + 24\cdot 31^{2} + 30\cdot 31^{3} + 20\cdot 31^{4} + 11\cdot 31^{5} + 4\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 + 16\cdot 31 + 8\cdot 31^{2} + 18\cdot 31^{4} + 28\cdot 31^{5} + 2\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 15 + 20\cdot 31 + 16\cdot 31^{2} + 20\cdot 31^{3} + 23\cdot 31^{4} + 17\cdot 31^{5} + 23\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 9 }$ | $=$ |
\( 16 + 10\cdot 31 + 14\cdot 31^{2} + 10\cdot 31^{3} + 7\cdot 31^{4} + 13\cdot 31^{5} + 7\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 10 }$ | $=$ |
\( 19 + 14\cdot 31 + 22\cdot 31^{2} + 30\cdot 31^{3} + 12\cdot 31^{4} + 2\cdot 31^{5} + 28\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 11 }$ | $=$ |
\( 21 + 6\cdot 31^{2} + 10\cdot 31^{4} + 19\cdot 31^{5} + 26\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 12 }$ | $=$ |
\( 22 + 26\cdot 31 + 19\cdot 31^{2} + 30\cdot 31^{3} + 7\cdot 31^{4} + 13\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 13 }$ | $=$ |
\( 23 + 28\cdot 31 + 3\cdot 31^{2} + 4\cdot 31^{3} + 12\cdot 31^{4} + 14\cdot 31^{5} + 14\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 14 }$ | $=$ |
\( 24 + 21\cdot 31 + 29\cdot 31^{2} + 8\cdot 31^{3} + 5\cdot 31^{4} + 27\cdot 31^{5} + 21\cdot 31^{6} +O(31^{7})\)
|
| $r_{ 15 }$ | $=$ |
\( 26 + 18\cdot 31 + 11\cdot 31^{2} + 22\cdot 31^{3} + 18\cdot 31^{4} + 11\cdot 31^{5} + 31^{6} +O(31^{7})\)
|
| $r_{ 16 }$ | $=$ |
\( 28 + 10\cdot 31 + 26\cdot 31^{2} + 26\cdot 31^{3} + 11\cdot 31^{4} + 9\cdot 31^{5} + 15\cdot 31^{6} +O(31^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 16 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 16 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)$ | $-1$ |
| $1$ | $4$ | $(1,14,16,3)(2,4,15,13)(5,8,12,9)(6,10,11,7)$ | $\zeta_{16}^{4}$ |
| $1$ | $4$ | $(1,3,16,14)(2,13,15,4)(5,9,12,8)(6,7,11,10)$ | $-\zeta_{16}^{4}$ |
| $1$ | $8$ | $(1,9,14,5,16,8,3,12)(2,11,4,7,15,6,13,10)$ | $-\zeta_{16}^{2}$ |
| $1$ | $8$ | $(1,5,3,9,16,12,14,8)(2,7,13,11,15,10,4,6)$ | $-\zeta_{16}^{6}$ |
| $1$ | $8$ | $(1,8,14,12,16,9,3,5)(2,6,4,10,15,11,13,7)$ | $\zeta_{16}^{2}$ |
| $1$ | $8$ | $(1,12,3,8,16,5,14,9)(2,10,13,6,15,7,4,11)$ | $\zeta_{16}^{6}$ |
| $1$ | $16$ | $(1,2,9,11,14,4,5,7,16,15,8,6,3,13,12,10)$ | $\zeta_{16}^{5}$ |
| $1$ | $16$ | $(1,11,5,15,3,10,9,4,16,6,12,2,14,7,8,13)$ | $-\zeta_{16}^{7}$ |
| $1$ | $16$ | $(1,4,8,10,14,15,12,11,16,13,9,7,3,2,5,6)$ | $-\zeta_{16}$ |
| $1$ | $16$ | $(1,7,12,4,3,11,8,2,16,10,5,13,14,6,9,15)$ | $\zeta_{16}^{3}$ |
| $1$ | $16$ | $(1,15,9,6,14,13,5,10,16,2,8,11,3,4,12,7)$ | $-\zeta_{16}^{5}$ |
| $1$ | $16$ | $(1,6,5,2,3,7,9,13,16,11,12,15,14,10,8,4)$ | $\zeta_{16}^{7}$ |
| $1$ | $16$ | $(1,13,8,7,14,2,12,6,16,4,9,10,3,15,5,11)$ | $\zeta_{16}$ |
| $1$ | $16$ | $(1,10,12,13,3,6,8,15,16,7,5,4,14,11,9,2)$ | $-\zeta_{16}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.