Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(182871529\)\(\medspace = 13523^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.13523.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.13523.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} - x^{2} - 3x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 433 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 32 + 9\cdot 433 + 89\cdot 433^{2} + 300\cdot 433^{3} + 336\cdot 433^{4} +O(433^{5})\) |
$r_{ 2 }$ | $=$ | \( 54 + 84\cdot 433 + 102\cdot 433^{2} + 23\cdot 433^{3} + 408\cdot 433^{4} +O(433^{5})\) |
$r_{ 3 }$ | $=$ | \( 66 + 308\cdot 433 + 192\cdot 433^{2} + 273\cdot 433^{3} + 122\cdot 433^{4} +O(433^{5})\) |
$r_{ 4 }$ | $=$ | \( 139 + 428\cdot 433 + 401\cdot 433^{2} + 176\cdot 433^{3} + 174\cdot 433^{4} +O(433^{5})\) |
$r_{ 5 }$ | $=$ | \( 143 + 36\cdot 433 + 80\cdot 433^{2} + 92\cdot 433^{3} + 257\cdot 433^{4} +O(433^{5})\) |
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 |
$1$ | $1$ | $()$ | $5$ |
$10$ | $2$ | $(1,2)$ | $1$ |
$15$ | $2$ | $(1,2)(3,4)$ | $1$ |
$20$ | $3$ | $(1,2,3)$ | $-1$ |
$30$ | $4$ | $(1,2,3,4)$ | $-1$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.