Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(15579283489\)\(\medspace = 7^{2} \cdot 11^{2} \cdot 1621^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.124817.1 |
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 5.5.124817.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 7x^{3} - 6x^{2} + 2x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 457 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 44 + 244\cdot 457 + 322\cdot 457^{2} + 104\cdot 457^{3} + 437\cdot 457^{4} +O(457^{5})\)
$r_{ 2 }$ |
$=$ |
\( 268 + 134\cdot 457 + 199\cdot 457^{2} + 181\cdot 457^{3} + 337\cdot 457^{4} +O(457^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 331 + 223\cdot 457 + 448\cdot 457^{2} + 121\cdot 457^{3} + 278\cdot 457^{4} +O(457^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 361 + 161\cdot 457 + 409\cdot 457^{2} + 40\cdot 457^{3} + 119\cdot 457^{4} +O(457^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 367 + 149\cdot 457 + 448\cdot 457^{2} + 7\cdot 457^{3} + 199\cdot 457^{4} +O(457^{5})\)
| |
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.