Properties

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

Related objects

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

 Size Order Action 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.