Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(3147076\)\(\medspace = 2^{2} \cdot 887^{2}\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | 5.1.3147076.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | 5.1.3147076.1 |
Defining polynomial
| $f(x)$ | $=$ | \(x^{5} - 2 x^{4} + 9 x^{3} - 12 x^{2} + 30 x + 2\)  . |
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | \( 6 + 38\cdot 181 + 181^{2} + 175\cdot 181^{3} + 124\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 2 }$ | $=$ | \( 11 + 31\cdot 181 + 125\cdot 181^{2} + 20\cdot 181^{3} + 137\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 3 }$ | $=$ | \( 41 + 166\cdot 181 + 120\cdot 181^{2} + 9\cdot 181^{3} + 10\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 4 }$ | $=$ | \( 143 + 82\cdot 181 + 88\cdot 181^{2} + 177\cdot 181^{3} + 103\cdot 181^{4} +O(181^{5})\)
|
| $r_{ 5 }$ | $=$ | \( 163 + 43\cdot 181 + 26\cdot 181^{2} + 160\cdot 181^{3} + 166\cdot 181^{4} +O(181^{5})\)
|
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.