Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(40000\)\(\medspace = 2^{6} \cdot 5^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.25000000.4 |
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 5.1.25000000.4 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 10x^{3} - 20x^{2} + 110x + 116 \) . |
The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 1 + 35\cdot 197 + 165\cdot 197^{2} + 58\cdot 197^{3} + 14\cdot 197^{4} +O(197^{5})\) |
$r_{ 2 }$ | $=$ | \( 85 + 40\cdot 197 + 153\cdot 197^{2} + 25\cdot 197^{3} + 182\cdot 197^{4} +O(197^{5})\) |
$r_{ 3 }$ | $=$ | \( 154 + 93\cdot 197 + 186\cdot 197^{2} + 155\cdot 197^{3} + 60\cdot 197^{4} +O(197^{5})\) |
$r_{ 4 }$ | $=$ | \( 164 + 150\cdot 197 + 104\cdot 197^{2} + 75\cdot 197^{3} + 110\cdot 197^{4} +O(197^{5})\) |
$r_{ 5 }$ | $=$ | \( 187 + 73\cdot 197 + 178\cdot 197^{2} + 77\cdot 197^{3} + 26\cdot 197^{4} +O(197^{5})\) |
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.