Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(34739908901584\)\(\medspace = 2^{4} \cdot 23^{3} \cdot 563^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.207184.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | even |
Determinant: | 1.12949.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.207184.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 6x^{3} + 2x^{2} + 7x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 13 + 163\cdot 283 + 163\cdot 283^{2} + 114\cdot 283^{3} + 118\cdot 283^{4} +O(283^{5})\) |
$r_{ 2 }$ | $=$ | \( 16 + 43\cdot 283 + 34\cdot 283^{2} + 11\cdot 283^{3} + 28\cdot 283^{4} +O(283^{5})\) |
$r_{ 3 }$ | $=$ | \( 57 + 189\cdot 283 + 126\cdot 283^{2} + 169\cdot 283^{3} + 185\cdot 283^{4} +O(283^{5})\) |
$r_{ 4 }$ | $=$ | \( 234 + 201\cdot 283 + 246\cdot 283^{2} + 142\cdot 283^{3} + 31\cdot 283^{4} +O(283^{5})\) |
$r_{ 5 }$ | $=$ | \( 247 + 251\cdot 283 + 277\cdot 283^{2} + 127\cdot 283^{3} + 202\cdot 283^{4} +O(283^{5})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$10$ | $2$ | $(1,2)$ | $0$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.