Properties

Label 3.17e2_29e2.12t33.1
Dimension 3
Group $A_5$
Conductor $ 17^{2} \cdot 29^{2}$
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$A_5$
Conductor:$243049= 17^{2} \cdot 29^{2} $
Artin number field: Splitting field of $f= x^{5} - x^{2} - 2 x - 3 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $A_5$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 56 + 147\cdot 157 + 77\cdot 157^{2} + 120\cdot 157^{3} + 109\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 81 + 28\cdot 157 + 70\cdot 157^{2} + 123\cdot 157^{3} + 99\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 106 + 32\cdot 157 + 140\cdot 157^{2} + 145\cdot 157^{3} + 69\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 111 + 90\cdot 157 + 151\cdot 157^{2} + 123\cdot 157^{3} + 90\cdot 157^{4} +O\left(157^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 117 + 14\cdot 157 + 31\cdot 157^{2} + 114\cdot 157^{3} + 100\cdot 157^{4} +O\left(157^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 5 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $3$ $3$
$15$ $2$ $(1,2)(3,4)$ $-1$ $-1$
$20$ $3$ $(1,2,3)$ $0$ $0$
$12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
$12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.