# Properties

 Label 3.193600.12t33.b.a Dimension 3 Group $A_5$ Conductor $2^{6} \cdot 5^{2} \cdot 11^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $193600= 2^{6} \cdot 5^{2} \cdot 11^{2}$ Artin number field: Splitting field of 5.1.193600.1 defined by $f= x^{5} + 2 x^{3} - 2 x^{2} - x + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective field: Galois closure of 5.1.193600.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 337 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $58 + 230\cdot 337 + 27\cdot 337^{2} + 118\cdot 337^{3} + 147\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 2 }$ $=$ $94 + 170\cdot 337 + 335\cdot 337^{2} + 55\cdot 337^{3} + 34\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 3 }$ $=$ $98 + 291\cdot 337 + 288\cdot 337^{2} + 320\cdot 337^{3} + 120\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 4 }$ $=$ $166 + 167\cdot 337 + 336\cdot 337^{2} + 293\cdot 337^{3} + 303\cdot 337^{4} +O\left(337^{ 5 }\right)$ $r_{ 5 }$ $=$ $258 + 151\cdot 337 + 22\cdot 337^{2} + 222\cdot 337^{3} + 67\cdot 337^{4} +O\left(337^{ 5 }\right)$

### 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 value $1$ $1$ $()$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.