Basic invariants
| Dimension: | $5$ |
| Group: | $A_5$ |
| Conductor: | \(12588304\)\(\medspace = 2^{4} \cdot 887^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.3147076.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $\PSL(2,5)$ |
| Parity: | even |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.3147076.1 |
Galois action
Roots of defining polynomial
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $0$ |