Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(53824\)\(\medspace = 2^{6} \cdot 29^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.53824.1 |
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 5.1.53824.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{3} - 2x^{2} + 3x + 2 \) . |
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 14 + 89\cdot 179 + 131\cdot 179^{2} + 60\cdot 179^{3} + 140\cdot 179^{4} +O(179^{5})\) |
$r_{ 2 }$ | $=$ | \( 79 + 139\cdot 179 + 86\cdot 179^{2} + 125\cdot 179^{3} + 43\cdot 179^{4} +O(179^{5})\) |
$r_{ 3 }$ | $=$ | \( 133 + 70\cdot 179 + 96\cdot 179^{2} + 66\cdot 179^{3} + 99\cdot 179^{4} +O(179^{5})\) |
$r_{ 4 }$ | $=$ | \( 153 + 81\cdot 179 + 77\cdot 179^{2} + 69\cdot 179^{3} + 155\cdot 179^{4} +O(179^{5})\) |
$r_{ 5 }$ | $=$ | \( 158 + 155\cdot 179 + 144\cdot 179^{2} + 35\cdot 179^{3} + 98\cdot 179^{4} +O(179^{5})\) |
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.