Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(193600\)\(\medspace = 2^{6} \cdot 5^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.193600.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $A_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $A_5$ |
| Projective stem field: | Galois closure of 5.1.193600.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + 2x^{3} - 2x^{2} - x + 2 \)
|
The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 58 + 230\cdot 337 + 27\cdot 337^{2} + 118\cdot 337^{3} + 147\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 94 + 170\cdot 337 + 335\cdot 337^{2} + 55\cdot 337^{3} + 34\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 98 + 291\cdot 337 + 288\cdot 337^{2} + 320\cdot 337^{3} + 120\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 166 + 167\cdot 337 + 336\cdot 337^{2} + 293\cdot 337^{3} + 303\cdot 337^{4} +O(337^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 258 + 151\cdot 337 + 22\cdot 337^{2} + 222\cdot 337^{3} + 67\cdot 337^{4} +O(337^{5})\)
|
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.