# Properties

 Label 3.2e6_5e2_11e2.12t33.2c2 Dimension 3 Group $A_5$ Conductor $2^{6} \cdot 5^{2} \cdot 11^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $193600= 2^{6} \cdot 5^{2} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{5} - 2 x^{4} + 6 x^{3} + 24 x^{2} + 37 x + 34$ 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 $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $18 a + 7 + \left(15 a + 17\right)\cdot 19 + \left(5 a + 12\right)\cdot 19^{2} + \left(7 a + 9\right)\cdot 19^{3} + \left(8 a + 9\right)\cdot 19^{4} + \left(6 a + 1\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ $r_{ 2 }$ $=$ $a + 6 + \left(3 a + 15\right)\cdot 19 + \left(13 a + 2\right)\cdot 19^{2} + \left(11 a + 11\right)\cdot 19^{3} + \left(10 a + 10\right)\cdot 19^{4} + \left(12 a + 18\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ $r_{ 3 }$ $=$ $18 a + 11 + 9\cdot 19 + \left(15 a + 11\right)\cdot 19^{2} + \left(17 a + 14\right)\cdot 19^{3} + \left(17 a + 12\right)\cdot 19^{4} + \left(17 a + 13\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$ $r_{ 4 }$ $=$ $6 + 3\cdot 19 + 4\cdot 19^{2} + 4\cdot 19^{3} + 11\cdot 19^{4} + 9\cdot 19^{5} +O\left(19^{ 6 }\right)$ $r_{ 5 }$ $=$ $a + 10 + \left(18 a + 11\right)\cdot 19 + \left(3 a + 6\right)\cdot 19^{2} + \left(a + 17\right)\cdot 19^{3} + \left(a + 12\right)\cdot 19^{4} + \left(a + 13\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$

### 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.