Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(13534019565191\)\(\medspace = 23831^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.3.23831.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | odd |
Determinant: | 1.23831.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.3.23831.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 3x^{3} - 2x^{2} + 4x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 31 + 5\cdot 223 + 175\cdot 223^{2} + 199\cdot 223^{3} + 120\cdot 223^{4} +O(223^{5})\) |
$r_{ 2 }$ | $=$ | \( 41 + 50\cdot 223 + 35\cdot 223^{2} + 203\cdot 223^{3} + 128\cdot 223^{4} +O(223^{5})\) |
$r_{ 3 }$ | $=$ | \( 76 + 139\cdot 223 + 124\cdot 223^{2} + 74\cdot 223^{3} + 184\cdot 223^{4} +O(223^{5})\) |
$r_{ 4 }$ | $=$ | \( 125 + 36\cdot 223 + 172\cdot 223^{2} + 102\cdot 223^{3} + 7\cdot 223^{4} +O(223^{5})\) |
$r_{ 5 }$ | $=$ | \( 173 + 214\cdot 223 + 161\cdot 223^{2} + 88\cdot 223^{3} + 4\cdot 223^{4} +O(223^{5})\) |
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.