Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(243049\)\(\medspace = 17^{2} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.243049.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.243049.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{2} - 2x - 3 \)
|
The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 56 + 147\cdot 157 + 77\cdot 157^{2} + 120\cdot 157^{3} + 109\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 81 + 28\cdot 157 + 70\cdot 157^{2} + 123\cdot 157^{3} + 99\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 106 + 32\cdot 157 + 140\cdot 157^{2} + 145\cdot 157^{3} + 69\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 111 + 90\cdot 157 + 151\cdot 157^{2} + 123\cdot 157^{3} + 90\cdot 157^{4} +O(157^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 117 + 14\cdot 157 + 31\cdot 157^{2} + 114\cdot 157^{3} + 100\cdot 157^{4} +O(157^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.