Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(4483962449\)\(\medspace = 17^{3} \cdot 97^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.1649.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | even |
Determinant: | 1.1649.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.1649.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + x^{2} - x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 499 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 121 + 359\cdot 499 + 431\cdot 499^{2} + 56\cdot 499^{3} + 376\cdot 499^{4} +O(499^{5})\)
$r_{ 2 }$ |
$=$ |
\( 123 + 266\cdot 499 + 309\cdot 499^{2} + 381\cdot 499^{3} + 84\cdot 499^{4} +O(499^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 163 + 125\cdot 499 + 219\cdot 499^{2} + 146\cdot 499^{3} + 173\cdot 499^{4} +O(499^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 180 + 69\cdot 499 + 489\cdot 499^{2} + 150\cdot 499^{3} +O(499^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 412 + 177\cdot 499 + 47\cdot 499^{2} + 262\cdot 499^{3} + 363\cdot 499^{4} +O(499^{5})\)
| |
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.