Basic invariants
| Dimension: | $5$ |
| Group: | $A_5$ |
| Conductor: | \(262440000\)\(\medspace = 2^{6} \cdot 3^{8} \cdot 5^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.129600.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PSL(2,5)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.129600.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{3} - 4x^{2} - 6x - 4 \)
|
The roots of $f$ are computed in $\Q_{ 467 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 49 + 50\cdot 467 + 28\cdot 467^{2} + 158\cdot 467^{3} + 139\cdot 467^{4} +O(467^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 59 + 367\cdot 467 + 148\cdot 467^{2} + 411\cdot 467^{3} + 177\cdot 467^{4} +O(467^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 102 + 228\cdot 467 + 80\cdot 467^{2} + 286\cdot 467^{3} + 364\cdot 467^{4} +O(467^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 319 + 331\cdot 467 + 385\cdot 467^{2} + 388\cdot 467^{3} + 209\cdot 467^{4} +O(467^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 405 + 423\cdot 467 + 290\cdot 467^{2} + 156\cdot 467^{3} + 42\cdot 467^{4} +O(467^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.