Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(287296\)\(\medspace = 2^{6} \cdot 67^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.287296.2 |
| 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 5.1.287296.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} + 2x^{3} - 4x^{2} + 6x - 4 \)
|
The roots of $f$ are computed in $\Q_{ 311 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 + 149\cdot 311 + 184\cdot 311^{2} + 99\cdot 311^{3} + 178\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 65 + 289\cdot 311 + 34\cdot 311^{2} + 249\cdot 311^{3} + 149\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 254 + 194\cdot 311 + 258\cdot 311^{2} + 273\cdot 311^{3} + 227\cdot 311^{4} +O(311^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 274 + 288\cdot 311 + 213\cdot 311^{2} + 283\cdot 311^{3} + 311^{4} +O(311^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 307 + 10\cdot 311 + 241\cdot 311^{2} + 26\cdot 311^{3} + 64\cdot 311^{4} +O(311^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.