# Properties

 Label 3.39204.12t33.a Dimension 3 Group $A_5$ Conductor $2^{2} \cdot 3^{4} \cdot 11^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $39204= 2^{2} \cdot 3^{4} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - 5 x^{3} + 13 x^{2} + 4 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.4743684.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots:
 $r_{ 1 }$ $=$ $26 a + 2 + \left(22 a + 29\right)\cdot 37 + \left(36 a + 15\right)\cdot 37^{2} + 27\cdot 37^{3} + \left(4 a + 14\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $31 + 32\cdot 37 + 32\cdot 37^{2} + 24\cdot 37^{3} + 8\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 a + 32 + \left(14 a + 19\right)\cdot 37 + 28\cdot 37^{2} + \left(36 a + 31\right)\cdot 37^{3} + \left(32 a + 29\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $12 a + \left(6 a + 8\right)\cdot 37 + \left(17 a + 4\right)\cdot 37^{2} + 29 a\cdot 37^{3} + \left(33 a + 13\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $25 a + 11 + \left(30 a + 21\right)\cdot 37 + \left(19 a + 29\right)\cdot 37^{2} + \left(7 a + 26\right)\cdot 37^{3} + \left(3 a + 7\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

### 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.