Basic invariants
| Dimension: | $1$ |
| Group: | $C_{15}$ |
| Conductor: | $341= 11 \cdot 31 $ |
| Artin number field: | Splitting field of $f= x^{15} - 2 x^{14} - 63 x^{13} + 133 x^{12} + 1422 x^{11} - 3182 x^{10} - 14085 x^{9} + 33507 x^{8} + 61557 x^{7} - 157405 x^{6} - 111611 x^{5} + 318268 x^{4} + 77338 x^{3} - 252062 x^{2} - 16054 x + 47081 $ over $\Q$ |
| Size of Galois orbit: | 8 |
| Smallest containing permutation representation: | $C_{15}$ |
| Parity: | Even |
| Corresponding Dirichlet character: | \(\chi_{341}(335,\cdot)\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{5} + 3 x + 27 $
Roots:
| $r_{ 1 }$ | $=$ | $ 5 a^{4} + 9 a^{3} + 3 a^{2} + 21 a + 11 + \left(21 a^{4} + 19 a^{3} + 9 a^{2} + 1\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 25 a^{2} + 2 a + 16\right)\cdot 29^{2} + \left(26 a^{4} + 24 a^{3} + 11 a^{2} + 9 a + 7\right)\cdot 29^{3} + \left(8 a^{4} + 28 a^{2} + 14 a + 8\right)\cdot 29^{4} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 19 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 5 a^{4} + 9 a^{3} + 3 a^{2} + 21 a + 13 + \left(21 a^{4} + 19 a^{3} + 9 a^{2} + 13\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 25 a^{2} + 2 a\right)\cdot 29^{2} + \left(26 a^{4} + 24 a^{3} + 11 a^{2} + 9 a + 20\right)\cdot 29^{3} + \left(8 a^{4} + 28 a^{2} + 14 a + 16\right)\cdot 29^{4} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 19 a + 11\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 5 a^{4} + 9 a^{3} + 3 a^{2} + 21 a + 24 + \left(21 a^{4} + 19 a^{3} + 9 a^{2} + 15\right)\cdot 29 + \left(4 a^{4} + 27 a^{3} + 25 a^{2} + 2 a + 17\right)\cdot 29^{2} + \left(26 a^{4} + 24 a^{3} + 11 a^{2} + 9 a + 21\right)\cdot 29^{3} + \left(8 a^{4} + 28 a^{2} + 14 a + 21\right)\cdot 29^{4} + \left(3 a^{4} + 15 a^{3} + 11 a^{2} + 19 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 7 a^{4} + 12 a^{3} + 13 a^{2} + 5 a + 10 + \left(5 a^{4} + 24 a^{3} + 19 a^{2} + 22 a + 15\right)\cdot 29 + \left(4 a^{4} + 28 a^{3} + 13 a^{2} + 5 a + 20\right)\cdot 29^{2} + \left(12 a^{4} + 28 a + 8\right)\cdot 29^{3} + \left(24 a^{4} + 13 a^{3} + 2 a^{2} + 22\right)\cdot 29^{4} + \left(23 a^{4} + 13 a^{3} + 22 a^{2} + 28 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 7 a^{4} + 12 a^{3} + 13 a^{2} + 5 a + 12 + \left(5 a^{4} + 24 a^{3} + 19 a^{2} + 22 a + 27\right)\cdot 29 + \left(4 a^{4} + 28 a^{3} + 13 a^{2} + 5 a + 4\right)\cdot 29^{2} + \left(12 a^{4} + 28 a + 21\right)\cdot 29^{3} + \left(24 a^{4} + 13 a^{3} + 2 a^{2} + 1\right)\cdot 29^{4} + \left(23 a^{4} + 13 a^{3} + 22 a^{2} + 28 a + 26\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 7 a^{4} + 12 a^{3} + 13 a^{2} + 5 a + 23 + \left(5 a^{4} + 24 a^{3} + 19 a^{2} + 22 a\right)\cdot 29 + \left(4 a^{4} + 28 a^{3} + 13 a^{2} + 5 a + 22\right)\cdot 29^{2} + \left(12 a^{4} + 28 a + 22\right)\cdot 29^{3} + \left(24 a^{4} + 13 a^{3} + 2 a^{2} + 6\right)\cdot 29^{4} + \left(23 a^{4} + 13 a^{3} + 22 a^{2} + 28 a + 16\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 9 a^{4} + 18 a^{3} + 6 a^{2} + 20 a + 9 + \left(24 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 26\right)\cdot 29 + \left(15 a^{4} + 23 a^{3} + 16 a^{2} + 27 a + 7\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + 20\right)\cdot 29^{3} + \left(21 a^{4} + 2 a^{3} + 20 a^{2} + 6 a + 3\right)\cdot 29^{4} + \left(28 a^{4} + 23 a^{3} + 15 a^{2} + 14 a + 25\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 9 a^{4} + 18 a^{3} + 6 a^{2} + 20 a + 11 + \left(24 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 9\right)\cdot 29 + \left(15 a^{4} + 23 a^{3} + 16 a^{2} + 27 a + 21\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + 3\right)\cdot 29^{3} + \left(21 a^{4} + 2 a^{3} + 20 a^{2} + 6 a + 12\right)\cdot 29^{4} + \left(28 a^{4} + 23 a^{3} + 15 a^{2} + 14 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 9 a^{4} + 18 a^{3} + 6 a^{2} + 20 a + 22 + \left(24 a^{4} + 5 a^{3} + 5 a^{2} + 10 a + 11\right)\cdot 29 + \left(15 a^{4} + 23 a^{3} + 16 a^{2} + 27 a + 9\right)\cdot 29^{2} + \left(14 a^{4} + 14 a^{3} + 23 a^{2} + 28 a + 5\right)\cdot 29^{3} + \left(21 a^{4} + 2 a^{3} + 20 a^{2} + 6 a + 17\right)\cdot 29^{4} + \left(28 a^{4} + 23 a^{3} + 15 a^{2} + 14 a + 10\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 13 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 7 + \left(9 a^{3} + 9 a^{2} + 25 a + 21\right)\cdot 29 + \left(28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 19\right)\cdot 29^{2} + \left(a^{4} + 25 a^{3} + 16 a^{2} + 12 a + 1\right)\cdot 29^{3} + \left(6 a^{4} + 11 a^{3} + 23 a^{2} + a + 13\right)\cdot 29^{4} + \left(2 a^{4} + 3 a^{3} + 15 a^{2} + 28 a + 19\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 11 }$ | $=$ | $ 13 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 9 + \left(9 a^{3} + 9 a^{2} + 25 a + 4\right)\cdot 29 + \left(28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 4\right)\cdot 29^{2} + \left(a^{4} + 25 a^{3} + 16 a^{2} + 12 a + 14\right)\cdot 29^{3} + \left(6 a^{4} + 11 a^{3} + 23 a^{2} + a + 21\right)\cdot 29^{4} + \left(2 a^{4} + 3 a^{3} + 15 a^{2} + 28 a + 14\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 12 }$ | $=$ | $ 13 a^{4} + 5 a^{3} + 2 a^{2} + 11 a + 20 + \left(9 a^{3} + 9 a^{2} + 25 a + 6\right)\cdot 29 + \left(28 a^{4} + 4 a^{3} + 27 a^{2} + 13 a + 21\right)\cdot 29^{2} + \left(a^{4} + 25 a^{3} + 16 a^{2} + 12 a + 15\right)\cdot 29^{3} + \left(6 a^{4} + 11 a^{3} + 23 a^{2} + a + 26\right)\cdot 29^{4} + \left(2 a^{4} + 3 a^{3} + 15 a^{2} + 28 a + 4\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 13 }$ | $=$ | $ 24 a^{4} + 14 a^{3} + 5 a^{2} + a + \left(6 a^{4} + 28 a^{3} + 15 a^{2} + 28 a + 22\right)\cdot 29 + \left(5 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 18\right)\cdot 29^{2} + \left(3 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 18\right)\cdot 29^{3} + \left(26 a^{4} + 12 a^{2} + 5 a + 16\right)\cdot 29^{4} + \left(28 a^{4} + 3 a^{3} + 21 a^{2} + 26 a + 22\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 14 }$ | $=$ | $ 24 a^{4} + 14 a^{3} + 5 a^{2} + a + 16 + \left(6 a^{4} + 28 a^{3} + 15 a^{2} + 28 a + 7\right)\cdot 29 + \left(5 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 17\right)\cdot 29^{2} + \left(3 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 4\right)\cdot 29^{3} + \left(26 a^{4} + 12 a^{2} + 5 a + 3\right)\cdot 29^{4} + \left(28 a^{4} + 3 a^{3} + 21 a^{2} + 26 a + 8\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$ |
| $r_{ 15 }$ | $=$ | $ 24 a^{4} + 14 a^{3} + 5 a^{2} + a + 18 + \left(6 a^{4} + 28 a^{3} + 15 a^{2} + 28 a + 19\right)\cdot 29 + \left(5 a^{4} + 2 a^{3} + 4 a^{2} + 8 a + 1\right)\cdot 29^{2} + \left(3 a^{4} + 21 a^{3} + 5 a^{2} + 8 a + 17\right)\cdot 29^{3} + \left(26 a^{4} + 12 a^{2} + 5 a + 11\right)\cdot 29^{4} + \left(28 a^{4} + 3 a^{3} + 21 a^{2} + 26 a + 3\right)\cdot 29^{5} +O\left(29^{ 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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)$ | $-\zeta_{15}^{5} - 1$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)$ | $\zeta_{15}^{5}$ |
| $1$ | $5$ | $(1,10,7,14,4)(2,11,8,15,5)(3,12,9,13,6)$ | $-\zeta_{15}^{7} - \zeta_{15}^{2}$ |
| $1$ | $5$ | $(1,7,4,10,14)(2,8,5,11,15)(3,9,6,12,13)$ | $\zeta_{15}^{7} - \zeta_{15}^{6} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
| $1$ | $5$ | $(1,14,10,4,7)(2,15,11,5,8)(3,13,12,6,9)$ | $\zeta_{15}^{6}$ |
| $1$ | $5$ | $(1,4,14,7,10)(2,5,15,8,11)(3,6,13,9,12)$ | $\zeta_{15}^{3}$ |
| $1$ | $15$ | $(1,11,9,14,5,3,10,8,13,4,2,12,7,15,6)$ | $\zeta_{15}^{7}$ |
| $1$ | $15$ | $(1,9,5,10,13,2,7,6,11,14,3,8,4,12,15)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ |
| $1$ | $15$ | $(1,5,13,7,11,3,4,15,9,10,2,6,14,8,12)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ |
| $1$ | $15$ | $(1,8,6,10,15,3,7,5,12,14,2,9,4,11,13)$ | $\zeta_{15}^{4}$ |
| $1$ | $15$ | $(1,13,11,4,9,2,14,12,5,7,3,15,10,6,8)$ | $-\zeta_{15}^{6} - \zeta_{15}$ |
| $1$ | $15$ | $(1,12,8,14,6,2,10,9,15,4,3,11,7,13,5)$ | $\zeta_{15}^{2}$ |
| $1$ | $15$ | $(1,15,12,4,8,3,14,11,6,7,2,13,10,5,9)$ | $\zeta_{15}$ |
| $1$ | $15$ | $(1,6,15,7,12,2,4,13,8,10,3,5,14,9,11)$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |