Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(1471182736\)\(\medspace = 2^{4} \cdot 43^{2} \cdot 223^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.153424.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.153424.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} - 4x^{3} + 8x^{2} - 2 \) . |
The roots of $f$ are computed in $\Q_{ 571 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 38 + 184\cdot 571 + 300\cdot 571^{2} + 297\cdot 571^{3} + 112\cdot 571^{4} +O(571^{5})\) |
$r_{ 2 }$ | $=$ | \( 358 + 364\cdot 571 + 173\cdot 571^{2} + 340\cdot 571^{3} + 203\cdot 571^{4} +O(571^{5})\) |
$r_{ 3 }$ | $=$ | \( 399 + 394\cdot 571 + 557\cdot 571^{2} + 363\cdot 571^{3} + 514\cdot 571^{4} +O(571^{5})\) |
$r_{ 4 }$ | $=$ | \( 412 + 212\cdot 571 + 558\cdot 571^{2} + 475\cdot 571^{3} + 185\cdot 571^{4} +O(571^{5})\) |
$r_{ 5 }$ | $=$ | \( 508 + 556\cdot 571 + 122\cdot 571^{2} + 235\cdot 571^{3} + 125\cdot 571^{4} +O(571^{5})\) |
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.