Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(449058481\)\(\medspace = 21191^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.21191.1 |
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 5.3.21191.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{2} - 3x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 112\cdot 257 + 39\cdot 257^{2} + 171\cdot 257^{3} + 199\cdot 257^{4} +O(257^{5})\) |
$r_{ 2 }$ | $=$ | \( 20 + 213\cdot 257 + 152\cdot 257^{2} + 58\cdot 257^{3} + 44\cdot 257^{4} +O(257^{5})\) |
$r_{ 3 }$ | $=$ | \( 55 + 106\cdot 257 + 180\cdot 257^{2} + 38\cdot 257^{3} + 134\cdot 257^{4} +O(257^{5})\) |
$r_{ 4 }$ | $=$ | \( 181 + 87\cdot 257 + 92\cdot 257^{2} + 133\cdot 257^{3} + 165\cdot 257^{4} +O(257^{5})\) |
$r_{ 5 }$ | $=$ | \( 253 + 251\cdot 257 + 48\cdot 257^{2} + 112\cdot 257^{3} + 227\cdot 257^{4} +O(257^{5})\) |
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.