Properties

Label 4.53_61.5t5.1
Dimension 4
Group $S_5$
Conductor $ 53 \cdot 61 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$3233= 53 \cdot 61 $
Artin number field: Splitting field of $f= x^{5} - x^{2} - 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 $\Q_{ 383 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 88 + 282\cdot 383 + 41\cdot 383^{2} + 170\cdot 383^{3} + 220\cdot 383^{4} +O\left(383^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 89 + 155\cdot 383 + 93\cdot 383^{2} + 241\cdot 383^{3} + 206\cdot 383^{4} +O\left(383^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 120 + 178\cdot 383 + 256\cdot 383^{2} + 144\cdot 383^{3} + 377\cdot 383^{4} +O\left(383^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 127 + 109\cdot 383 + 48\cdot 383^{2} + 292\cdot 383^{3} + 181\cdot 383^{4} +O\left(383^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 342 + 40\cdot 383 + 326\cdot 383^{2} + 300\cdot 383^{3} + 162\cdot 383^{4} +O\left(383^{ 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.