Basic invariants
| Dimension: | $4$ |
| Group: | $A_5$ |
| Conductor: | \(245025\)\(\medspace = 3^{4} \cdot 5^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.245025.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.245025.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 3x^{3} - 4x^{2} + 6x + 3 \)
|
The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 + 103\cdot 131 + 80\cdot 131^{2} + 69\cdot 131^{3} + 64\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 94\cdot 131 + 7\cdot 131^{2} + 46\cdot 131^{3} + 44\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 99 + 106\cdot 131 + 69\cdot 131^{2} + 68\cdot 131^{3} + 131^{4} +O(131^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 116 + 37\cdot 131 + 33\cdot 131^{2} + 24\cdot 131^{3} + 43\cdot 131^{4} +O(131^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 127 + 50\cdot 131 + 70\cdot 131^{2} + 53\cdot 131^{3} + 108\cdot 131^{4} +O(131^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.