Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(56073665725\)\(\medspace = 5^{2} \cdot 7^{3} \cdot 11^{3} \cdot 17^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.32725.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.1309.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.32725.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} + 2x^{3} + 2x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 5 + 67\cdot 97 + 16\cdot 97^{3} + 10\cdot 97^{4} +O(97^{5})\) |
$r_{ 2 }$ | $=$ | \( 40 + 73\cdot 97 + 7\cdot 97^{2} + 20\cdot 97^{3} + 89\cdot 97^{4} +O(97^{5})\) |
$r_{ 3 }$ | $=$ | \( 69 + 11\cdot 97 + 38\cdot 97^{2} + 62\cdot 97^{3} + 47\cdot 97^{4} +O(97^{5})\) |
$r_{ 4 }$ | $=$ | \( 89 + 18\cdot 97 + 51\cdot 97^{2} + 48\cdot 97^{3} + 29\cdot 97^{4} +O(97^{5})\) |
$r_{ 5 }$ | $=$ | \( 90 + 22\cdot 97 + 96\cdot 97^{2} + 46\cdot 97^{3} + 17\cdot 97^{4} +O(97^{5})\) |
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.