Basic invariants
| Dimension: | $3$ |
| Group: | $A_5$ |
| Conductor: | \(77841\)\(\medspace = 3^{4} \cdot 31^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.1.74805201.1 |
| 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 5.1.74805201.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - x^{4} + 19x^{3} - 41x^{2} + 32x - 5 \)
|
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 37 + 102\cdot 389 + 312\cdot 389^{2} + 343\cdot 389^{3} + 14\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 + 385\cdot 389 + 204\cdot 389^{2} + 157\cdot 389^{3} + 111\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 72 + 135\cdot 389 + 59\cdot 389^{2} + 47\cdot 389^{3} + 249\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 86 + 332\cdot 389 + 226\cdot 389^{2} + 362\cdot 389^{3} + 331\cdot 389^{4} +O(389^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 150 + 212\cdot 389 + 363\cdot 389^{2} + 255\cdot 389^{3} + 70\cdot 389^{4} +O(389^{5})\)
|
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.