Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(2635849611859456\)\(\medspace = 2^{9} \cdot 31^{3} \cdot 557^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.138136.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.138136.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.138136.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 6x^{3} + 3x^{2} + 4x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 443 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 65 + 366\cdot 443 + 206\cdot 443^{2} + 193\cdot 443^{3} + 240\cdot 443^{4} +O(443^{5})\) |
$r_{ 2 }$ | $=$ | \( 86 + 6\cdot 443 + 121\cdot 443^{2} + 222\cdot 443^{4} +O(443^{5})\) |
$r_{ 3 }$ | $=$ | \( 157 + 215\cdot 443 + 325\cdot 443^{2} + 66\cdot 443^{3} + 392\cdot 443^{4} +O(443^{5})\) |
$r_{ 4 }$ | $=$ | \( 203 + 158\cdot 443 + 430\cdot 443^{2} + 105\cdot 443^{3} + 27\cdot 443^{4} +O(443^{5})\) |
$r_{ 5 }$ | $=$ | \( 376 + 139\cdot 443 + 245\cdot 443^{2} + 76\cdot 443^{3} + 4\cdot 443^{4} +O(443^{5})\) |
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.