Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(646541456023\)\(\medspace = 8647^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.8647.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | odd |
Determinant: | 1.8647.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.8647.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 3x^{3} - 2x^{2} + 2x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 563 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 33 + 208\cdot 563 + 332\cdot 563^{2} + 389\cdot 563^{3} + 74\cdot 563^{4} +O(563^{5})\)
$r_{ 2 }$ |
$=$ |
\( 360 + 91\cdot 563 + 108\cdot 563^{2} + 190\cdot 563^{3} + 235\cdot 563^{4} +O(563^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 363 + 532\cdot 563 + 519\cdot 563^{2} + 379\cdot 563^{3} + 19\cdot 563^{4} +O(563^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 434 + 492\cdot 563 + 53\cdot 563^{2} + 293\cdot 563^{3} + 409\cdot 563^{4} +O(563^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 499 + 363\cdot 563 + 111\cdot 563^{2} + 436\cdot 563^{3} + 386\cdot 563^{4} +O(563^{5})\)
| |
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.