# Properties

 Label 3.50625.12t33.b.b Dimension 3 Group $A_5$ Conductor $3^{4} \cdot 5^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $50625= 3^{4} \cdot 5^{4}$ Artin number field: Splitting field of 5.1.31640625.1 defined by $f= x^{5} - 25 x^{2} + 75$ 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 5.1.31640625.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $22 + 5\cdot 71 + 2\cdot 71^{2} + 46\cdot 71^{3} + 41\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 2 }$ $=$ $25 + 55\cdot 71 + 64\cdot 71^{2} + 5\cdot 71^{3} + 13\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 3 }$ $=$ $36 + 70\cdot 71 + 18\cdot 71^{2} + 57\cdot 71^{3} + 31\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 4 }$ $=$ $63 + 44\cdot 71 + 58\cdot 71^{2} + 67\cdot 71^{3} + 42\cdot 71^{4} +O\left(71^{ 5 }\right)$ $r_{ 5 }$ $=$ $67 + 36\cdot 71 + 68\cdot 71^{2} + 35\cdot 71^{3} + 12\cdot 71^{4} +O\left(71^{ 5 }\right)$

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