Basic invariants
Dimension: | $4$ |
Group: | $S_5$ |
Conductor: | \(126032\)\(\medspace = 2^{4} \cdot 7877 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.126032.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $S_5$ |
Parity: | even |
Determinant: | 1.7877.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.126032.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - 6x^{3} + 6x - 2 \) . |
The roots of $f$ are computed in $\Q_{ 431 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 41 + 283\cdot 431 + 194\cdot 431^{2} + 263\cdot 431^{3} + 86\cdot 431^{4} +O(431^{5})\)
$r_{ 2 }$ |
$=$ |
\( 237 + 167\cdot 431 + 161\cdot 431^{2} + 102\cdot 431^{3} + 76\cdot 431^{4} +O(431^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 305 + 397\cdot 431 + 274\cdot 431^{2} + 383\cdot 431^{3} + 337\cdot 431^{4} +O(431^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 349 + 335\cdot 431 + 238\cdot 431^{2} + 243\cdot 431^{3} + 178\cdot 431^{4} +O(431^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 361 + 108\cdot 431 + 423\cdot 431^{2} + 299\cdot 431^{3} + 182\cdot 431^{4} +O(431^{5})\)
| |
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.