Properties

Label 4.4261.5t5.1
Dimension 4
Group $S_5$
Conductor $ 4261 $
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$S_5$
Conductor:$4261 $
Artin number field: Splitting field of $f= x^{5} - 2 x^{2} - 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 an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 11 + 12\cdot 17 + 2\cdot 17^{2} + 12\cdot 17^{3} + 14\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 a + 1 + \left(4 a + 8\right)\cdot 17 + \left(10 a + 10\right)\cdot 17^{2} + \left(11 a + 4\right)\cdot 17^{3} + \left(a + 16\right)\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 8 + 3\cdot 17 + 15\cdot 17^{2} + 11\cdot 17^{3} + 4\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 7 + 3\cdot 17 + 6\cdot 17^{2} + 16\cdot 17^{3} + 8\cdot 17^{4} +O\left(17^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 7 + \left(12 a + 6\right)\cdot 17 + \left(6 a + 16\right)\cdot 17^{2} + \left(5 a + 5\right)\cdot 17^{3} + \left(15 a + 6\right)\cdot 17^{4} +O\left(17^{ 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.