Basic invariants
Dimension: | $5$ |
Group: | $S_5$ |
Conductor: | \(41134735489\)\(\medspace = 202817^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.202817.1 |
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 5.5.202817.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 2x^{4} - 4x^{3} + 5x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 347 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 6 + 27\cdot 347 + 187\cdot 347^{2} + 57\cdot 347^{3} + 183\cdot 347^{4} +O(347^{5})\) |
$r_{ 2 }$ | $=$ | \( 131 + 215\cdot 347 + 18\cdot 347^{2} + 339\cdot 347^{3} + 234\cdot 347^{4} +O(347^{5})\) |
$r_{ 3 }$ | $=$ | \( 178 + 36\cdot 347 + 130\cdot 347^{2} + 188\cdot 347^{3} + 279\cdot 347^{4} +O(347^{5})\) |
$r_{ 4 }$ | $=$ | \( 190 + 168\cdot 347 + 193\cdot 347^{2} + 36\cdot 347^{3} + 18\cdot 347^{4} +O(347^{5})\) |
$r_{ 5 }$ | $=$ | \( 191 + 246\cdot 347 + 164\cdot 347^{2} + 72\cdot 347^{3} + 325\cdot 347^{4} +O(347^{5})\) |
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.