# Properties

 Label 3.570025.12t33.a.b Dimension $3$ Group $A_5$ Conductor $570025$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $$570025$$$$\medspace = 5^{2} \cdot 151^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 5.1.570025.1 Galois orbit size: $2$ Smallest permutation container: $A_5$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_5$ Projective stem field: 5.1.570025.1

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - 3 x^{3} - x^{2} + x - 3$$  .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$4 a + 4 + 8\cdot 11 + \left(7 a + 4\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(6 a + 5\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 2 }$ $=$ $$7 a + 9 + \left(10 a + 5\right)\cdot 11 + \left(3 a + 10\right)\cdot 11^{2} + 9\cdot 11^{3} + 4 a\cdot 11^{4} +O(11^{5})$$ $r_{ 3 }$ $=$ $$a + 5 + \left(3 a + 9\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(6 a + 10\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 4 }$ $=$ $$6 + 10\cdot 11 + 7\cdot 11^{2} + 9\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})$$ $r_{ 5 }$ $=$ $$10 a + 9 + \left(7 a + 9\right)\cdot 11 + \left(3 a + 6\right)\cdot 11^{2} + 9\cdot 11^{3} + \left(4 a + 5\right)\cdot 11^{4} +O(11^{5})$$

## 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.