# Properties

 Label 3.105625.12t33.a Dimension 3 Group $A_5$ Conductor $5^{4} \cdot 13^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $105625= 5^{4} \cdot 13^{2}$ Artin number field: Splitting field of $f= x^{5} + 5 x^{3} - 10 x^{2} - 45$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.2640625.1

## 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:
 $r_{ 1 }$ $=$ $16 + 34\cdot 37^{2} + 11\cdot 37^{3} + 15\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 2 }$ $=$ $23 a + 29 + \left(4 a + 30\right)\cdot 37 + \left(6 a + 24\right)\cdot 37^{2} + \left(7 a + 3\right)\cdot 37^{3} + \left(32 a + 22\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 3 }$ $=$ $14 a + 10 + \left(32 a + 26\right)\cdot 37 + \left(30 a + 7\right)\cdot 37^{2} + \left(29 a + 26\right)\cdot 37^{3} + \left(4 a + 32\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 4 }$ $=$ $14 a + \left(9 a + 15\right)\cdot 37 + \left(35 a + 30\right)\cdot 37^{2} + \left(12 a + 7\right)\cdot 37^{3} + \left(3 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$ $r_{ 5 }$ $=$ $23 a + 19 + \left(27 a + 1\right)\cdot 37 + \left(a + 14\right)\cdot 37^{2} + \left(24 a + 24\right)\cdot 37^{3} + \left(33 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

### 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.