Basic invariants
| Dimension: | $2$ |
| Group: | $D_5\times C_5$ |
| Conductor: | $15125= 5^{3} \cdot 11^{2} $ |
| Artin number field: | Splitting field of $f= x^{10} - 15 x^{8} - 10 x^{7} + 55 x^{6} + 53 x^{5} - 40 x^{4} - 50 x^{3} - 5 x^{2} + 5 x + 1 $ over $\Q$ |
| Size of Galois orbit: | 4 |
| Smallest containing permutation representation: | $D_5\times C_5$ |
| Parity: | Even |
| Determinant: | 1.5_11.10t1.1c1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{5} + 3 x + 27 $
Roots:
| $r_{ 1 }$ | $=$ | $ 2 a^{4} + 2 a^{3} + 5 a^{2} + 24 a + 20 + \left(4 a^{4} + 17 a^{3} + 7 a^{2} + 3 a + 18\right)\cdot 29 + \left(20 a^{4} + 22 a^{3} + 2 a^{2} + 6 a + 11\right)\cdot 29^{2} + \left(3 a^{4} + 16 a^{3} + 3 a^{2} + 13 a + 3\right)\cdot 29^{3} + \left(28 a^{4} + 7 a^{3} + 23 a^{2} + 27 a + 15\right)\cdot 29^{4} + \left(21 a^{4} + 23 a^{3} + 22 a^{2} + 10 a + 11\right)\cdot 29^{5} + \left(15 a^{4} + 24 a^{3} + 22 a^{2} + 15 a + 3\right)\cdot 29^{6} + \left(a^{4} + 26 a^{3} + 10 a^{2} + 4 a + 21\right)\cdot 29^{7} + \left(24 a^{4} + 26 a^{3} + 2 a + 2\right)\cdot 29^{8} + \left(12 a^{4} + 3 a^{3} + 21 a^{2} + 3 a + 27\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 2 }$ | $=$ | $ 2 a^{4} + 28 a^{3} + 3 a^{2} + 6 a + 7 + \left(21 a^{4} + 26 a^{3} + 22 a^{2} + 24\right)\cdot 29 + \left(a^{4} + a^{3} + 2 a^{2} + 25 a + 11\right)\cdot 29^{2} + \left(28 a^{4} + 13 a^{3} + 8 a^{2} + 22 a + 26\right)\cdot 29^{3} + \left(9 a^{4} + 15 a^{3} + 6 a^{2} + 16 a\right)\cdot 29^{4} + \left(6 a^{4} + 5 a^{3} + 12 a^{2} + 28 a + 10\right)\cdot 29^{5} + \left(4 a^{4} + 3 a^{3} + 22 a^{2} + 6 a + 21\right)\cdot 29^{6} + \left(a^{4} + 4 a^{3} + 5 a^{2} + 11 a + 2\right)\cdot 29^{7} + \left(3 a^{4} + 5 a^{3} + 27 a^{2} + 18 a + 10\right)\cdot 29^{8} + \left(21 a^{4} + 15 a^{3} + 18 a^{2} + 28 a + 25\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 3 }$ | $=$ | $ 3 a^{4} + 16 a^{3} + 13 a^{2} + 25 a + 21 + \left(28 a^{4} + 11 a^{3} + 23 a^{2} + 18 a + 23\right)\cdot 29 + \left(9 a^{4} + 6 a^{3} + 16 a^{2} + 5 a + 25\right)\cdot 29^{2} + \left(8 a^{4} + 19 a^{3} + 26 a^{2} + 10 a + 13\right)\cdot 29^{3} + \left(24 a^{4} + 23 a^{3} + 25 a^{2} + 12 a\right)\cdot 29^{4} + \left(13 a^{4} + 11 a^{3} + 20 a^{2} + 4 a + 28\right)\cdot 29^{5} + \left(24 a^{4} + 3 a^{3} + 26 a^{2} + 16 a + 11\right)\cdot 29^{6} + \left(28 a^{4} + 22 a^{3} + 21 a^{2} + 24 a + 5\right)\cdot 29^{7} + \left(6 a^{4} + 4 a^{3} + 17 a^{2} + 5 a + 2\right)\cdot 29^{8} + \left(a^{4} + 6 a^{3} + 21 a^{2} + 7 a + 24\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 4 }$ | $=$ | $ 6 a^{4} + 23 a^{3} + 22 a^{2} + 28 a + 18 + \left(27 a^{4} + 16 a^{3} + 11 a^{2} + 8 a + 4\right)\cdot 29 + \left(8 a^{4} + 19 a^{3} + 12 a^{2} + 22 a + 2\right)\cdot 29^{2} + \left(4 a^{4} + 26 a^{3} + 18 a^{2} + 23 a + 28\right)\cdot 29^{3} + \left(2 a^{4} + 9 a^{3} + 13 a^{2} + 11 a + 4\right)\cdot 29^{4} + \left(10 a^{4} + 6 a^{3} + 14 a^{2} + 5 a + 6\right)\cdot 29^{5} + \left(5 a^{4} + 6 a^{3} + 3 a^{2} + 21 a + 13\right)\cdot 29^{6} + \left(19 a^{4} + 4 a^{3} + a^{2} + 13 a + 5\right)\cdot 29^{7} + \left(12 a^{4} + 3 a^{3} + 23 a^{2} + 13 a + 10\right)\cdot 29^{8} + \left(15 a^{4} + 11 a^{3} + 20 a^{2} + 7 a + 4\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 5 }$ | $=$ | $ 11 a^{4} + 23 a^{3} + 7 a^{2} + 24 a + 17 + \left(19 a^{4} + 24 a^{3} + 9 a + 8\right)\cdot 29 + \left(13 a + 9\right)\cdot 29^{2} + \left(7 a^{4} + 18 a^{3} + 4 a^{2} + 24 a + 22\right)\cdot 29^{3} + \left(9 a^{4} + 22 a^{3} + a^{2} + 20 a + 10\right)\cdot 29^{4} + \left(6 a^{4} + 15 a^{3} + 26 a^{2} + 10 a + 27\right)\cdot 29^{5} + \left(26 a^{4} + 20 a^{3} + 4 a^{2} + 8 a + 21\right)\cdot 29^{6} + \left(6 a^{4} + 21 a^{3} + a^{2} + 19 a + 10\right)\cdot 29^{7} + \left(20 a^{4} + 27 a^{3} + 23 a^{2} + 11 a + 22\right)\cdot 29^{8} + \left(7 a^{4} + 18 a^{3} + 21 a^{2} + 24 a + 4\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 6 }$ | $=$ | $ 13 a^{4} + 24 a^{3} + 25 a^{2} + 27 a + \left(13 a^{4} + 23 a^{2} + 17 a + 18\right)\cdot 29 + \left(5 a^{4} + 26 a^{3} + 18 a^{2} + a + 28\right)\cdot 29^{2} + \left(23 a^{4} + 6 a^{3} + 20 a^{2} + 23 a + 26\right)\cdot 29^{3} + \left(21 a^{4} + 20 a^{3} + 24 a^{2} + 20 a + 28\right)\cdot 29^{4} + \left(10 a^{4} + 21 a^{3} + 13 a^{2} + 18 a + 1\right)\cdot 29^{5} + \left(14 a^{4} + 6 a^{3} + 2 a\right)\cdot 29^{6} + \left(a^{4} + 5 a^{2} + 10 a + 21\right)\cdot 29^{7} + \left(23 a^{4} + 27 a^{3} + 23 a^{2} + 16 a + 17\right)\cdot 29^{8} + \left(18 a^{4} + 9 a^{3} + 2 a^{2} + 18 a + 6\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 7 }$ | $=$ | $ 14 a^{4} + 13 a^{3} + 7 a^{2} + a + 1 + \left(15 a^{4} + 11 a^{3} + 7 a^{2} + 25 a + 5\right)\cdot 29 + \left(3 a^{4} + 23 a^{3} + 5 a^{2} + 2 a + 16\right)\cdot 29^{2} + \left(28 a^{4} + 26 a^{3} + 12 a^{2} + 5 a + 26\right)\cdot 29^{3} + \left(19 a^{4} + 28 a^{3} + 26 a^{2} + 19 a + 24\right)\cdot 29^{4} + \left(24 a^{4} + 17 a^{3} + 25 a^{2} + 20 a + 1\right)\cdot 29^{5} + \left(3 a^{4} + 4 a^{3} + 15 a^{2} + 22 a + 3\right)\cdot 29^{6} + \left(15 a^{4} + 5 a^{3} + 21 a^{2} + a + 13\right)\cdot 29^{7} + \left(17 a^{4} + 13 a^{3} + 23 a^{2} + 2 a + 27\right)\cdot 29^{8} + \left(26 a^{4} + 20 a^{3} + 10 a^{2} + 26\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 8 }$ | $=$ | $ 16 a^{4} + 24 a^{3} + 15 a^{2} + 22 a + 13 + \left(8 a^{4} + 8 a^{3} + 18 a^{2} + 13 a\right)\cdot 29 + \left(19 a^{4} + 27 a^{3} + 15 a^{2} + 3 a + 27\right)\cdot 29^{2} + \left(3 a^{4} + 25 a^{3} + 6 a^{2} + 22 a + 14\right)\cdot 29^{3} + \left(26 a^{4} + 26 a^{3} + 12 a^{2} + 4 a + 4\right)\cdot 29^{4} + \left(28 a^{4} + 10 a^{3} + 6 a^{2} + a + 28\right)\cdot 29^{5} + \left(a^{4} + 2 a^{3} + 9 a^{2} + 6 a + 10\right)\cdot 29^{6} + \left(25 a^{4} + 21 a^{3} + 6 a^{2} + 6 a + 25\right)\cdot 29^{7} + \left(21 a^{4} + 23 a^{3} + 27 a^{2} + 23 a + 14\right)\cdot 29^{8} + \left(24 a^{4} + 4 a^{3} + 18 a^{2} + 19 a + 3\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 9 }$ | $=$ | $ 21 a^{4} + 14 a^{3} + 20 a^{2} + 15 a + 25 + \left(4 a^{4} + 14 a^{3} + 25 a^{2} + 13 a + 2\right)\cdot 29 + \left(4 a^{4} + 20 a^{3} + 8 a^{2} + 24 a + 8\right)\cdot 29^{2} + \left(23 a^{4} + 10 a^{3} + 9 a^{2} + 4 a + 15\right)\cdot 29^{3} + \left(8 a^{4} + 22 a^{3} + 13 a^{2} + 22 a + 3\right)\cdot 29^{4} + \left(15 a^{4} + 24 a^{3} + 21 a + 7\right)\cdot 29^{5} + \left(20 a^{4} + 17 a^{3} + 22 a^{2} + 12 a + 3\right)\cdot 29^{6} + \left(10 a^{4} + 5 a^{3} + 5 a^{2} + 23 a + 14\right)\cdot 29^{7} + \left(5 a^{4} + 6 a^{3} + 13 a^{2} + 2 a + 27\right)\cdot 29^{8} + \left(15 a^{4} + 28 a^{3} + 23 a^{2} + 9 a + 26\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
| $r_{ 10 }$ | $=$ | $ 28 a^{4} + 7 a^{3} + 28 a^{2} + 2 a + 23 + \left(2 a^{4} + 12 a^{3} + 4 a^{2} + 4 a + 9\right)\cdot 29 + \left(13 a^{4} + 25 a^{3} + 4 a^{2} + 11 a + 4\right)\cdot 29^{2} + \left(15 a^{4} + 9 a^{3} + 7 a^{2} + 24 a + 25\right)\cdot 29^{3} + \left(23 a^{4} + 25 a^{3} + 27 a^{2} + 17 a + 21\right)\cdot 29^{4} + \left(6 a^{4} + 6 a^{3} + a^{2} + 22 a + 22\right)\cdot 29^{5} + \left(28 a^{4} + 26 a^{3} + 17 a^{2} + 3 a + 26\right)\cdot 29^{6} + \left(5 a^{4} + 4 a^{3} + 7 a^{2} + a + 25\right)\cdot 29^{7} + \left(10 a^{4} + 7 a^{3} + 24 a^{2} + 20 a + 9\right)\cdot 29^{8} + \left(a^{4} + 26 a^{3} + 13 a^{2} + 26 a + 24\right)\cdot 29^{9} +O\left(29^{ 10 }\right)$ |
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 value |
| $1$ | $1$ | $()$ | $2$ |
| $5$ | $2$ | $(1,10)(2,8)(3,4)(5,9)(6,7)$ | $0$ |
| $1$ | $5$ | $(1,6,9,4,8)(2,10,7,5,3)$ | $2 \zeta_{5}$ |
| $1$ | $5$ | $(1,9,8,6,4)(2,7,3,10,5)$ | $2 \zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,4,6,8,9)(2,5,10,3,7)$ | $2 \zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,8,4,9,6)(2,3,5,7,10)$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ |
| $2$ | $5$ | $(2,5,10,3,7)$ | $\zeta_{5}^{2} + \zeta_{5}$ |
| $2$ | $5$ | $(2,10,7,5,3)$ | $-\zeta_{5}^{3} - \zeta_{5} - 1$ |
| $2$ | $5$ | $(2,3,5,7,10)$ | $\zeta_{5}^{3} + \zeta_{5}$ |
| $2$ | $5$ | $(2,7,3,10,5)$ | $-\zeta_{5}^{2} - \zeta_{5} - 1$ |
| $2$ | $5$ | $(1,8,4,9,6)(2,7,3,10,5)$ | $\zeta_{5} + 1$ |
| $2$ | $5$ | $(1,4,6,8,9)(2,3,5,7,10)$ | $\zeta_{5}^{2} + 1$ |
| $2$ | $5$ | $(1,9,8,6,4)(2,10,7,5,3)$ | $\zeta_{5}^{3} + 1$ |
| $2$ | $5$ | $(1,6,9,4,8)(2,5,10,3,7)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5}$ |
| $2$ | $5$ | $(1,6,9,4,8)(2,3,5,7,10)$ | $\zeta_{5}^{3} + \zeta_{5}^{2}$ |
| $2$ | $5$ | $(1,9,8,6,4)(2,5,10,3,7)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - 1$ |
| $5$ | $10$ | $(1,3,6,2,9,10,4,7,8,5)$ | $0$ |
| $5$ | $10$ | $(1,2,4,5,6,10,8,3,9,7)$ | $0$ |
| $5$ | $10$ | $(1,7,9,3,8,10,6,5,4,2)$ | $0$ |
| $5$ | $10$ | $(1,5,8,7,4,10,9,2,6,3)$ | $0$ |