# Properties

 Label 3.261121.12t33.a Dimension $3$ Group $A_5$ Conductor $261121$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $$261121$$$$\medspace = 7^{2} \cdot 73^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 5.1.261121.1 Galois orbit size: $2$ Smallest permutation container: $A_5$ Parity: even Projective image: $A_5$ Projective field: Galois closure of 5.1.261121.1

## 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:
 $r_{ 1 }$ $=$ $a + 3 + \left(22 a + 14\right)\cdot 23 + \left(22 a + 8\right)\cdot 23^{2} + \left(4 a + 18\right)\cdot 23^{3} + \left(12 a + 10\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $2 a + 21 + \left(5 a + 4\right)\cdot 23 + \left(18 a + 20\right)\cdot 23^{2} + \left(6 a + 20\right)\cdot 23^{3} + \left(a + 2\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $15 + 2\cdot 23 + 2\cdot 23^{2} + 8\cdot 23^{3} + 3\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $22 a + 5 + 11\cdot 23 + 9\cdot 23^{2} + \left(18 a + 5\right)\cdot 23^{3} + \left(10 a + 7\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $21 a + 2 + \left(17 a + 13\right)\cdot 23 + \left(4 a + 5\right)\cdot 23^{2} + \left(16 a + 16\right)\cdot 23^{3} + \left(21 a + 21\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

### 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 values $c1$ $c2$ $1$ $1$ $()$ $3$ $3$ $15$ $2$ $(1,2)(3,4)$ $-1$ $-1$ $20$ $3$ $(1,2,3)$ $0$ $0$ $12$ $5$ $(1,2,3,4,5)$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $12$ $5$ $(1,3,4,5,2)$ $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ $-\zeta_{5}^{3} - \zeta_{5}^{2}$
The blue line marks the conjugacy class containing complex conjugation.