Basic invariants
| Dimension: | $1$ |
| Group: | $C_{15}$ |
| Conductor: | $99= 3^{2} \cdot 11 $ |
| Artin number field: | Splitting field of $f= x^{15} - 27 x^{13} - 4 x^{12} + 252 x^{11} + 60 x^{10} - 976 x^{9} - 288 x^{8} + 1473 x^{7} + 384 x^{6} - 765 x^{5} - 168 x^{4} + 150 x^{3} + 27 x^{2} - 9 x - 1 $ over $\Q$ |
| Size of Galois orbit: | 8 |
| Smallest containing permutation representation: | $C_{15}$ |
| Parity: | Even |
| Corresponding Dirichlet character: | \(\chi_{99}(31,\cdot)\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{5} + x + 14 $
Roots:
| $r_{ 1 }$ | $=$ | $ a^{4} + 14 a^{3} + 15 a^{2} + 14 a + 13 + \left(10 a^{4} + 11 a^{3} + 14 a^{2} + 10 a + 3\right)\cdot 17 + \left(11 a^{4} + 8 a^{3} + 15 a^{2} + 13 a\right)\cdot 17^{2} + \left(2 a^{4} + 8 a^{3} + 7 a^{2} + 6 a + 14\right)\cdot 17^{3} + \left(14 a^{4} + 4 a^{3} + 11 a^{2} + 5 a + 3\right)\cdot 17^{4} + \left(a^{4} + 2 a^{3} + 16 a^{2} + 16 a + 15\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 2 a^{4} + 11 a^{3} + 13 a^{2} + 11 a + 9 + \left(15 a^{3} + a^{2} + 13 a + 2\right)\cdot 17 + \left(9 a^{4} + a^{3} + 6 a^{2} + 14 a + 10\right)\cdot 17^{2} + \left(7 a^{4} + 8 a^{3} + 12 a^{2} + 7 a + 5\right)\cdot 17^{3} + \left(13 a^{4} + 13 a^{3} + 3 a^{2} + 2 a + 8\right)\cdot 17^{4} + \left(13 a^{4} + 2 a^{3} + 7 a^{2} + 16 a + 13\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 3 a^{4} + 14 a^{3} + 8 a^{2} + 9 a + 1 + \left(3 a^{4} + 8 a^{3} + 6 a^{2} + 11 a + 5\right)\cdot 17 + \left(2 a^{4} + 10 a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 17^{2} + \left(15 a^{4} + 16 a^{3} + 14 a^{2} + 2 a + 10\right)\cdot 17^{3} + \left(7 a^{4} + 2 a^{3} + a + 5\right)\cdot 17^{4} + \left(16 a^{4} + 14 a^{3} + 9 a^{2} + 7 a + 13\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 4 a^{4} + 8 a^{3} + 5 a^{2} + 16 a + 12 + \left(16 a^{4} + 12 a^{3} + 9 a^{2} + 16 a + 8\right)\cdot 17 + \left(12 a^{4} + 3 a^{3} + 7 a^{2} + 12 a + 11\right)\cdot 17^{2} + \left(14 a^{3} + 6 a^{2} + 8 a + 5\right)\cdot 17^{3} + \left(8 a^{4} + 15 a^{3} + 4 a^{2} + 5 a + 2\right)\cdot 17^{4} + \left(15 a^{4} + 6 a^{3} + a^{2} + 10 a + 9\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 5 a^{4} + 10 a^{3} + 2 a^{2} + 3 a + 15 + \left(14 a^{4} + 3 a^{3} + 4 a^{2} + 14 a + 9\right)\cdot 17 + \left(13 a^{4} + 10 a^{2} + 10 a + 3\right)\cdot 17^{2} + \left(13 a^{3} + 15 a^{2} + 8 a + 6\right)\cdot 17^{3} + \left(12 a^{4} + 11 a^{3} + 12 a^{2} + 16 a + 9\right)\cdot 17^{4} + \left(6 a^{4} + 7 a^{3} + 5 a^{2} + 3 a + 9\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 6 a^{4} + 11 a^{3} + 16 a^{2} + a + 2 + \left(14 a^{4} + 9 a^{3} + 5 a^{2} + 13 a + 7\right)\cdot 17 + \left(8 a^{4} + 14 a^{3} + 13 a + 13\right)\cdot 17^{2} + \left(8 a^{4} + 4 a^{3} + 8 a^{2} + 7 a + 9\right)\cdot 17^{3} + \left(5 a^{4} + 13 a^{3} + 15 a^{2} + a + 15\right)\cdot 17^{4} + \left(10 a^{4} + 16 a^{3} + 12 a^{2} + 7 a + 3\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 7 a^{4} + 5 a^{3} + 9 a^{2} + 13 + \left(13 a^{4} + 8 a^{3} + 8 a^{2} + 5 a + 9\right)\cdot 17 + \left(6 a^{4} + 16 a^{3} + 2 a^{2} + 6 a + 1\right)\cdot 17^{2} + \left(15 a^{4} + 3 a^{3} + 6 a^{2} + a + 5\right)\cdot 17^{3} + \left(5 a^{4} + 13 a^{3} + 5 a^{2} + 2 a + 2\right)\cdot 17^{4} + \left(9 a^{4} + 13 a^{3} + 4 a^{2} + 5 a + 3\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 8 a^{4} + 9 a^{3} + 10 a^{2} + 7 a + 14 + \left(16 a^{4} + 15 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 17 + \left(5 a^{4} + 8 a^{3} + 11 a^{2} + 8 a + 14\right)\cdot 17^{2} + \left(10 a^{4} + 12 a^{3} + 11 a^{2} + 6 a + 13\right)\cdot 17^{3} + \left(3 a^{4} + 14 a + 12\right)\cdot 17^{4} + \left(7 a^{4} + 3 a^{3} + 12 a^{2} + 2 a + 16\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 8 a^{4} + 16 a^{3} + 10 a^{2} + 15 a + 7 + \left(3 a^{4} + 3 a^{2} + 2 a + 15\right)\cdot 17 + \left(7 a^{4} + 13 a^{3} + 16 a^{2} + 10 a + 1\right)\cdot 17^{2} + \left(15 a^{4} + 6 a^{3} + 11 a^{2} + 16 a + 5\right)\cdot 17^{3} + \left(13 a^{4} + 6 a^{3} + 16 a^{2} + 11 a + 5\right)\cdot 17^{4} + \left(11 a^{4} + 2 a^{3} + 9 a^{2} + 2 a + 15\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 9 a^{4} + 5 a^{3} + 4 a^{2} + 15 a + 8 + \left(3 a^{3} + 6 a^{2} + 16 a + 12\right)\cdot 17 + \left(3 a^{4} + 3 a^{3} + 12 a^{2} + 1\right)\cdot 17^{2} + \left(9 a^{4} + 7 a^{3} + 11 a^{2} + 11 a + 6\right)\cdot 17^{3} + \left(2 a^{4} + 8 a^{3} + a + 15\right)\cdot 17^{4} + \left(a^{4} + 5 a^{3} + 7 a^{2} + 11\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 11 a^{4} + 8 a^{3} + 3 a^{2} + 7 a + 6 + \left(2 a^{4} + 16 a^{3} + 14 a^{2} + 16 a + 11\right)\cdot 17 + \left(2 a^{4} + 4 a^{3} + 8 a^{2} + 5 a + 4\right)\cdot 17^{2} + \left(4 a^{4} + 10 a^{3} + 12 a^{2} + 6\right)\cdot 17^{3} + \left(12 a^{4} + 4 a^{3} + 9 a^{2} + 16 a + 7\right)\cdot 17^{4} + \left(5 a^{4} + 15 a^{3} + 16 a^{2} + 2 a + 10\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 12 a^{4} + 11 a^{3} + 13 a^{2} + 15 + \left(7 a^{4} + 3 a^{3} + 6 a^{2} + 11 a + 1\right)\cdot 17 + \left(12 a^{4} + 2 a^{3} + 9 a^{2} + 2 a + 11\right)\cdot 17^{2} + \left(11 a^{4} + 12 a^{3} + 12 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(a^{4} + 6 a^{3} + 10 a^{2} + 5 a + 7\right)\cdot 17^{4} + \left(7 a^{4} + 14 a^{3} + 13 a^{2} + 3 a + 2\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 13 }$ | $=$ | $ 14 a^{4} + 4 a^{3} + 10 a^{2} + 12 a + 3 + \left(13 a^{4} + 14 a^{3} + 13 a^{2} + 10\right)\cdot 17 + \left(11 a^{4} + 8 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 17^{2} + \left(3 a^{4} + 16 a^{3} + 9 a^{2} + 5 a + 4\right)\cdot 17^{3} + \left(2 a^{4} + 3 a^{3} + 6 a^{2} + 16 a + 11\right)\cdot 17^{4} + \left(10 a^{4} + 13 a^{3} + 10 a^{2} + 13 a + 11\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 14 }$ | $=$ | $ 14 a^{4} + 9 a^{3} + 6 a^{2} + 9 a + 12 + \left(6 a^{4} + 6 a^{3} + 9 a + 10\right)\cdot 17 + \left(13 a^{4} + 6 a^{3} + 12 a^{2} + 5 a + 6\right)\cdot 17^{2} + \left(6 a^{4} + 13 a^{2} + 2 a + 14\right)\cdot 17^{3} + \left(6 a^{4} + 16 a^{3} + a^{2} + 9 a + 4\right)\cdot 17^{4} + \left(a^{4} + 11 a^{3} + 10 a^{2} + a + 5\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
| $r_{ 15 }$ | $=$ | $ 15 a^{4} + a^{3} + 12 a^{2} + 6 + \left(12 a^{4} + 5 a^{3} + a^{2} + a + 5\right)\cdot 17 + \left(14 a^{4} + 15 a^{3} + 5 a^{2} + 8 a + 4\right)\cdot 17^{2} + \left(6 a^{4} + 15 a^{2} + 5 a + 4\right)\cdot 17^{3} + \left(9 a^{4} + 14 a^{3} + 9 a + 7\right)\cdot 17^{4} + \left(5 a^{3} + 16 a^{2} + 8 a + 11\right)\cdot 17^{5} +O\left(17^{ 6 }\right)$ |
Generators of the action on the roots $r_1, \ldots, r_{ 15 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 15 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,14)(3,6,8)(4,9,5)(7,15,12)(10,13,11)$ | $-\zeta_{15}^{5} - 1$ |
| $1$ | $3$ | $(1,14,2)(3,8,6)(4,5,9)(7,12,15)(10,11,13)$ | $\zeta_{15}^{5}$ |
| $1$ | $5$ | $(1,3,4,12,13)(2,6,9,7,11)(5,15,10,14,8)$ | $\zeta_{15}^{6}$ |
| $1$ | $5$ | $(1,4,13,3,12)(2,9,11,6,7)(5,10,8,15,14)$ | $-\zeta_{15}^{7} - \zeta_{15}^{2}$ |
| $1$ | $5$ | $(1,12,3,13,4)(2,7,6,11,9)(5,14,15,8,10)$ | $\zeta_{15}^{3}$ |
| $1$ | $5$ | $(1,13,12,4,3)(2,11,7,9,6)(5,8,14,10,15)$ | $\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
| $1$ | $15$ | $(1,6,5,12,11,14,3,9,15,13,2,8,4,7,10)$ | $\zeta_{15}$ |
| $1$ | $15$ | $(1,5,11,3,15,2,4,10,6,12,14,9,13,8,7)$ | $\zeta_{15}^{2}$ |
| $1$ | $15$ | $(1,11,15,4,6,14,13,7,5,3,2,10,12,9,8)$ | $\zeta_{15}^{4}$ |
| $1$ | $15$ | $(1,9,10,3,7,14,4,11,8,12,2,5,13,6,15)$ | $\zeta_{15}^{7}$ |
| $1$ | $15$ | $(1,15,6,13,5,2,12,8,11,4,14,7,3,10,9)$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
| $1$ | $15$ | $(1,8,9,12,10,2,3,5,7,13,14,6,4,15,11)$ | $-\zeta_{15}^{6} - \zeta_{15}$ |
| $1$ | $15$ | $(1,7,8,13,9,14,12,6,10,4,2,15,3,11,5)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ |
| $1$ | $15$ | $(1,10,7,4,8,2,13,15,9,3,14,11,12,5,6)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ |