Properties

Label 4.17_97.5t5.1
Dimension 4
Group $S_5$
Conductor $ 17 \cdot 97 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$1649= 17 \cdot 97 $
Artin number field: Splitting field of $f= x^{5} - x^{4} + x^{2} - x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_5$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 499 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 121 + 359\cdot 499 + 431\cdot 499^{2} + 56\cdot 499^{3} + 376\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 123 + 266\cdot 499 + 309\cdot 499^{2} + 381\cdot 499^{3} + 84\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 163 + 125\cdot 499 + 219\cdot 499^{2} + 146\cdot 499^{3} + 173\cdot 499^{4} +O\left(499^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 180 + 69\cdot 499 + 489\cdot 499^{2} + 150\cdot 499^{3} +O\left(499^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 412 + 177\cdot 499 + 47\cdot 499^{2} + 262\cdot 499^{3} + 363\cdot 499^{4} +O\left(499^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

Cycle notation
$(1,2)$
$(1,2,3,4,5)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$
$1$ $1$ $()$ $4$
$10$ $2$ $(1,2)$ $2$
$15$ $2$ $(1,2)(3,4)$ $0$
$20$ $3$ $(1,2,3)$ $1$
$30$ $4$ $(1,2,3,4)$ $0$
$24$ $5$ $(1,2,3,4,5)$ $-1$
$20$ $6$ $(1,2,3)(4,5)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.