# Properties

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

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $40000= 2^{6} \cdot 5^{4}$ Artin number field: Splitting field of $f= x^{5} + 20 x - 16$ 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 $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $2 a + 18 + \left(4 a + 11\right)\cdot 23 + \left(7 a + 19\right)\cdot 23^{2} + \left(14 a + 18\right)\cdot 23^{3} + 21 a\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $12 a + 17 + \left(21 a + 6\right)\cdot 23 + \left(19 a + 17\right)\cdot 23^{2} + \left(16 a + 16\right)\cdot 23^{3} + \left(22 a + 17\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $21 a + 22 + \left(18 a + 17\right)\cdot 23 + \left(15 a + 6\right)\cdot 23^{2} + \left(8 a + 17\right)\cdot 23^{3} + \left(a + 6\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $17 + 17\cdot 23 + 12\cdot 23^{2} + 8\cdot 23^{3} + 20\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 18 + \left(a + 14\right)\cdot 23 + \left(3 a + 12\right)\cdot 23^{2} + \left(6 a + 7\right)\cdot 23^{3} +O\left(23^{ 5 }\right)$

### 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.