Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(137707850944\)\(\medspace = 2^{6} \cdot 1291^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.5164.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $\PGL(2,5)$ |
Parity: | even |
Determinant: | 1.5164.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.5164.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - x^{3} + 2x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 57 + 67\cdot 283 + 157\cdot 283^{2} + 9\cdot 283^{3} + 61\cdot 283^{4} +O(283^{5})\) |
$r_{ 2 }$ | $=$ | \( 154 + 11\cdot 283 + 124\cdot 283^{2} + 196\cdot 283^{3} + 198\cdot 283^{4} +O(283^{5})\) |
$r_{ 3 }$ | $=$ | \( 180 + 118\cdot 283 + 256\cdot 283^{2} + 131\cdot 283^{3} + 99\cdot 283^{4} +O(283^{5})\) |
$r_{ 4 }$ | $=$ | \( 198 + 254\cdot 283 + 20\cdot 283^{2} + 86\cdot 283^{3} + 137\cdot 283^{4} +O(283^{5})\) |
$r_{ 5 }$ | $=$ | \( 261 + 113\cdot 283 + 7\cdot 283^{2} + 142\cdot 283^{3} + 69\cdot 283^{4} +O(283^{5})\) |
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.