Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(383448334625\)\(\medspace = 5^{3} \cdot 1453^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.1.7265.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.7265.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.1.7265.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} + x^{3} + 2x^{2} - x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 503 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 30 + 249\cdot 503 + 298\cdot 503^{2} + 57\cdot 503^{3} + 191\cdot 503^{4} +O(503^{5})\) |
$r_{ 2 }$ | $=$ | \( 52 + 215\cdot 503 + 470\cdot 503^{2} + 361\cdot 503^{3} + 60\cdot 503^{4} +O(503^{5})\) |
$r_{ 3 }$ | $=$ | \( 295 + 274\cdot 503 + 186\cdot 503^{2} + 344\cdot 503^{3} + 258\cdot 503^{4} +O(503^{5})\) |
$r_{ 4 }$ | $=$ | \( 297 + 118\cdot 503 + 197\cdot 503^{2} + 380\cdot 503^{3} + 84\cdot 503^{4} +O(503^{5})\) |
$r_{ 5 }$ | $=$ | \( 333 + 148\cdot 503 + 356\cdot 503^{2} + 364\cdot 503^{3} + 410\cdot 503^{4} +O(503^{5})\) |
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.