Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(13799\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.13799.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | odd |
Determinant: | 1.13799.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.13799.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} + 2x^{2} - 3x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 9 + 49\cdot 97 + 11\cdot 97^{2} + 74\cdot 97^{3} + 15\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 16 + 60\cdot 97 + 4\cdot 97^{2} + 48\cdot 97^{3} + 18\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 48 + 83\cdot 97 + 17\cdot 97^{2} + 51\cdot 97^{3} + 78\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 59 + 74\cdot 97 + 11\cdot 97^{2} + 51\cdot 97^{3} + 86\cdot 97^{4} +O(97^{5})\) |
$r_{ 5 }$ | $=$ | \( 63 + 23\cdot 97 + 51\cdot 97^{2} + 66\cdot 97^{3} + 91\cdot 97^{4} +O(97^{5})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$10$ | $2$ | $(1,2)$ | $2$ |
$15$ | $2$ | $(1,2)(3,4)$ | $0$ |
$20$ | $3$ | $(1,2,3)$ | $1$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.