# Properties

 Label 3.3e2_79e2.12t33.1c1 Dimension 3 Group $A_5$ Conductor $3^{2} \cdot 79^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $56169= 3^{2} \cdot 79^{2}$ Artin number field: Splitting field of $f= x^{5} - 79 x^{2} + 474 x - 711$ 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 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $11 a + 7 + \left(22 a + 15\right)\cdot 23 + \left(12 a + 7\right)\cdot 23^{2} + \left(14 a + 3\right)\cdot 23^{3} + \left(15 a + 13\right)\cdot 23^{4} + \left(15 a + 18\right)\cdot 23^{5} + \left(12 a + 20\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 2 }$ $=$ $12 a + 6 + 3\cdot 23 + \left(10 a + 11\right)\cdot 23^{2} + \left(8 a + 19\right)\cdot 23^{3} + \left(7 a + 6\right)\cdot 23^{4} + \left(7 a + 11\right)\cdot 23^{5} + \left(10 a + 7\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 3 }$ $=$ $8 a + 9 + 16\cdot 23 + \left(17 a + 16\right)\cdot 23^{2} + \left(20 a + 3\right)\cdot 23^{3} + \left(19 a + 16\right)\cdot 23^{4} + 22 a\cdot 23^{5} + 12\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 4 }$ $=$ $22 + 23 + 6\cdot 23^{2} + 14\cdot 23^{3} + 20\cdot 23^{4} + 11\cdot 23^{5} + 14\cdot 23^{6} +O\left(23^{ 7 }\right)$ $r_{ 5 }$ $=$ $15 a + 2 + \left(22 a + 9\right)\cdot 23 + \left(5 a + 4\right)\cdot 23^{2} + \left(2 a + 5\right)\cdot 23^{3} + \left(3 a + 12\right)\cdot 23^{4} + 3\cdot 23^{5} + \left(22 a + 14\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$

### 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.