Basic invariants
Dimension: | $3$ |
Group: | $A_5$ |
Conductor: | \(50625\)\(\medspace = 3^{4} \cdot 5^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.31640625.1 |
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 5.1.31640625.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 25x^{2} + 75 \) . |
The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 22 + 5\cdot 71 + 2\cdot 71^{2} + 46\cdot 71^{3} + 41\cdot 71^{4} +O(71^{5})\)
$r_{ 2 }$ |
$=$ |
\( 25 + 55\cdot 71 + 64\cdot 71^{2} + 5\cdot 71^{3} + 13\cdot 71^{4} +O(71^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 36 + 70\cdot 71 + 18\cdot 71^{2} + 57\cdot 71^{3} + 31\cdot 71^{4} +O(71^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 63 + 44\cdot 71 + 58\cdot 71^{2} + 67\cdot 71^{3} + 42\cdot 71^{4} +O(71^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 67 + 36\cdot 71 + 68\cdot 71^{2} + 35\cdot 71^{3} + 12\cdot 71^{4} +O(71^{5})\)
| |
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.