Basic invariants
Dimension: | $6$ |
Group: | $S_5$ |
Conductor: | \(13236200869377149\)\(\medspace = 236549^{3} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 5.5.236549.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 20T30 |
Parity: | even |
Determinant: | 1.236549.2t1.a.a |
Projective image: | $S_5$ |
Projective stem field: | Galois closure of 5.5.236549.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{5} - x^{4} - 6x^{3} + 7x^{2} + 2x - 1 \) . |
The roots of $f$ are computed in $\Q_{ 461 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 98 + 99\cdot 461 + 277\cdot 461^{2} + 443\cdot 461^{3} + 317\cdot 461^{4} +O(461^{5})\)
$r_{ 2 }$ |
$=$ |
\( 203 + 279\cdot 461 + 175\cdot 461^{2} + 315\cdot 461^{3} + 430\cdot 461^{4} +O(461^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 247 + 346\cdot 461 + 218\cdot 461^{2} + 382\cdot 461^{3} + 138\cdot 461^{4} +O(461^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 386 + 441\cdot 461 + 9\cdot 461^{2} + 144\cdot 461^{3} + 38\cdot 461^{4} +O(461^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 450 + 215\cdot 461 + 240\cdot 461^{2} + 97\cdot 461^{3} + 457\cdot 461^{4} +O(461^{5})\)
| |
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 |
$1$ | $1$ | $()$ | $6$ |
$10$ | $2$ | $(1,2)$ | $0$ |
$15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
$20$ | $3$ | $(1,2,3)$ | $0$ |
$30$ | $4$ | $(1,2,3,4)$ | $0$ |
$24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
$20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.