Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(6494855411\)\(\medspace = 11^{3} \cdot 47^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.24299.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | odd |
Determinant: | 1.11.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.24299.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} + 3x^{2} - x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 12 + 45\cdot 97 + 5\cdot 97^{2} + 69\cdot 97^{3} + 59\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 47 + 66\cdot 97 + 53\cdot 97^{2} + 73\cdot 97^{3} + 24\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 74 + 91\cdot 97 + 23\cdot 97^{2} + 38\cdot 97^{3} + 70\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 79 + 47\cdot 97 + 69\cdot 97^{2} + 53\cdot 97^{3} + 55\cdot 97^{4} +O(97^{5})\) |
$r_{ 5 }$ | $=$ | \( 80 + 39\cdot 97 + 41\cdot 97^{2} + 56\cdot 97^{3} + 80\cdot 97^{4} +O(97^{5})\) |
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.