Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(126025\)\(\medspace = 5^{2} \cdot 71^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.126025.1 |
| 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 5.1.126025.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 3x^{3} - 4x^{2} + 5x - 1 \)
|
The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 232 + 411\cdot 457 + 331\cdot 457^{2} + 218\cdot 457^{3} + 88\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 265 + 412\cdot 457 + 390\cdot 457^{2} + 376\cdot 457^{3} + 347\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 434 + 441\cdot 457 + 117\cdot 457^{2} + 448\cdot 457^{3} + 125\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 444 + 338\cdot 457 + 281\cdot 457^{2} + 19\cdot 457^{3} + 138\cdot 457^{4} +O(457^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 454 + 222\cdot 457 + 248\cdot 457^{2} + 307\cdot 457^{3} + 213\cdot 457^{4} +O(457^{5})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ | ✓ |
| $20$ | $3$ | $(1,2,3)$ | $0$ | |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ | |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |