# Properties

 Label 3.40000.12t33.c.a 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 5.1.25000000.3 defined by $f= x^{5} + 10 x^{3} - 40 x^{2} + 60 x - 32$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective field: Galois closure of 5.1.25000000.3

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $x^{2} + 49 x + 2$
Roots:
 $r_{ 1 }$ $=$ $6 a + 10 + \left(24 a + 21\right)\cdot 53 + \left(24 a + 20\right)\cdot 53^{2} + \left(8 a + 51\right)\cdot 53^{3} + \left(41 a + 30\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 2 }$ $=$ $23 a + 39 + \left(29 a + 19\right)\cdot 53 + \left(12 a + 24\right)\cdot 53^{2} + \left(21 a + 8\right)\cdot 53^{3} + \left(39 a + 36\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 3 }$ $=$ $30 a + 25 + \left(23 a + 8\right)\cdot 53 + \left(40 a + 45\right)\cdot 53^{2} + \left(31 a + 27\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 4 }$ $=$ $51 + 50\cdot 53 + 27\cdot 53^{2} + 10\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$ $r_{ 5 }$ $=$ $47 a + 34 + \left(28 a + 5\right)\cdot 53 + \left(28 a + 41\right)\cdot 53^{2} + \left(44 a + 7\right)\cdot 53^{3} + \left(11 a + 28\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$

### 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 $1$ $1$ $()$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$
The blue line marks the conjugacy class containing complex conjugation.