Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(6293711178847\)\(\medspace = 37^{3} \cdot 499^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.18463.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | odd |
Determinant: | 1.18463.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.18463.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 3x^{3} + 4x^{2} + x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 16 + 32\cdot 61 + 58\cdot 61^{2} + 21\cdot 61^{3} + 27\cdot 61^{4} +O(61^{5})\) |
$r_{ 2 }$ | $=$ | \( 32 + 2\cdot 61 + 22\cdot 61^{2} + 4\cdot 61^{3} + 46\cdot 61^{4} +O(61^{5})\) |
$r_{ 3 }$ | $=$ | \( 39 + 29\cdot 61 + 55\cdot 61^{2} + 32\cdot 61^{3} + 17\cdot 61^{4} +O(61^{5})\) |
$r_{ 4 }$ | $=$ | \( 47 + 8\cdot 61 + 32\cdot 61^{2} + 12\cdot 61^{3} + 6\cdot 61^{4} +O(61^{5})\) |
$r_{ 5 }$ | $=$ | \( 50 + 48\cdot 61 + 14\cdot 61^{2} + 50\cdot 61^{3} + 24\cdot 61^{4} +O(61^{5})\) |
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.