Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(948676\)\(\medspace = 2^{2} \cdot 487^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.948676.1 |
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 5.1.948676.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} + 5x^{3} + 10x^{2} + 2 \) . |
The roots of $f$ are computed in $\Q_{ 199 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 62 + 152\cdot 199 + 105\cdot 199^{2} + 96\cdot 199^{3} + 3\cdot 199^{4} +O(199^{5})\) |
$r_{ 2 }$ | $=$ | \( 104 + 186\cdot 199 + 28\cdot 199^{2} + 26\cdot 199^{3} + 77\cdot 199^{4} +O(199^{5})\) |
$r_{ 3 }$ | $=$ | \( 125 + 135\cdot 199 + 172\cdot 199^{2} + 153\cdot 199^{3} + 190\cdot 199^{4} +O(199^{5})\) |
$r_{ 4 }$ | $=$ | \( 133 + 96\cdot 199 + 119\cdot 199^{2} + 97\cdot 199^{3} + 198\cdot 199^{4} +O(199^{5})\) |
$r_{ 5 }$ | $=$ | \( 175 + 25\cdot 199 + 170\cdot 199^{2} + 23\cdot 199^{3} + 127\cdot 199^{4} +O(199^{5})\) |
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.