# Properties

 Label 3.2e6_5e4.12t33.2c2 Dimension 3 Group $A_5$ Conductor $2^{6} \cdot 5^{4}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $40000= 2^{6} \cdot 5^{4}$ Artin number field: Splitting field of $f= x^{5} + 10 x^{3} - 10 x^{2} + 35 x - 18$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Determinant: 1.1.1t1.1c1

## Galois action

### Roots of defining polynomial

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

### 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.