# Properties

 Label 3.70225.12t33.a Dimension 3 Group $A_5$ Conductor $5^{2} \cdot 53^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $70225= 5^{2} \cdot 53^{2}$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} - x^{3} + 3 x^{2} - x + 2$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.70225.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $13 a + 10 + \left(4 a + 10\right)\cdot 19 + \left(17 a + 4\right)\cdot 19^{2} + \left(6 a + 2\right)\cdot 19^{3} + \left(2 a + 7\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $4 + 6\cdot 19 + 17\cdot 19^{2} + 3\cdot 19^{3} + 13\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $8 a + 7 + 13\cdot 19 + \left(4 a + 16\right)\cdot 19^{2} + 4 a\cdot 19^{3} + \left(18 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $6 a + 4 + \left(14 a + 2\right)\cdot 19 + \left(a + 17\right)\cdot 19^{2} + \left(12 a + 10\right)\cdot 19^{3} + \left(16 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 15 + \left(18 a + 5\right)\cdot 19 + \left(14 a + 1\right)\cdot 19^{2} + \left(14 a + 1\right)\cdot 19^{3} + 5\cdot 19^{4} +O\left(19^{ 5 }\right)$

### 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.