Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(61504\)\(\medspace = 2^{6} \cdot 31^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.59105344.1 |
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 5.1.59105344.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} + 14x^{3} - 18x^{2} + 47x - 36 \) . |
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 54 + 21\cdot 257 + 142\cdot 257^{2} + 138\cdot 257^{3} + 153\cdot 257^{4} +O(257^{5})\)
$r_{ 2 }$ |
$=$ |
\( 90 + 102\cdot 257 + 212\cdot 257^{2} + 217\cdot 257^{3} + 210\cdot 257^{4} +O(257^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 107 + 6\cdot 257 + 204\cdot 257^{2} + 210\cdot 257^{3} + 181\cdot 257^{4} +O(257^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 117 + 113\cdot 257 + 152\cdot 257^{2} + 253\cdot 257^{3} + 223\cdot 257^{4} +O(257^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 148 + 13\cdot 257 + 60\cdot 257^{2} + 207\cdot 257^{3} +O(257^{5})\)
| |
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.