Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(4338889\)\(\medspace = 2083^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 5.1.4338889.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Projective image: | $A_5$ |
| Projective field: | Galois closure of 5.1.4338889.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 241 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 47 + 194\cdot 241 + 5\cdot 241^{2} + 200\cdot 241^{3} + 188\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 88 + 84\cdot 241 + 45\cdot 241^{2} + 31\cdot 241^{3} + 45\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 175 + 27\cdot 241 + 12\cdot 241^{2} + 65\cdot 241^{3} + 218\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 188 + 88\cdot 241 + 233\cdot 241^{2} + 186\cdot 241^{3} + 5\cdot 241^{4} +O(241^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 226 + 86\cdot 241 + 185\cdot 241^{2} + 239\cdot 241^{3} + 23\cdot 241^{4} +O(241^{5})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $3$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |