# Properties

 Label 3.88209.12t33.b Dimension 3 Group $A_5$ Conductor $3^{6} \cdot 11^{2}$ Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_5$ Conductor: $88209= 3^{6} \cdot 11^{2}$ Artin number field: Splitting field of $f= x^{5} - x^{4} + 7 x^{3} + 3 x^{2} - 30 x + 21$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $A_5$ Parity: Even Projective image: $A_5$ Projective field: Galois closure of 5.1.10673289.2

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $x^{2} + 6 x + 3$
Roots:
 $r_{ 1 }$ $=$ $5\cdot 7 + 6\cdot 7^{2} + 4\cdot 7^{3} + 7^{5} + 3\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 2 }$ $=$ $a + \left(6 a + 4\right)\cdot 7 + \left(5 a + 5\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(6 a + 5\right)\cdot 7^{5} + \left(5 a + 1\right)\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 3 }$ $=$ $2 a + 6 + 3 a\cdot 7 + \left(5 a + 4\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(4 a + 2\right)\cdot 7^{4} + 5 a\cdot 7^{5} + \left(5 a + 4\right)\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 4 }$ $=$ $6 a + 1 + 2\cdot 7 + \left(a + 5\right)\cdot 7^{2} + \left(a + 2\right)\cdot 7^{3} + \left(a + 1\right)\cdot 7^{4} + 6\cdot 7^{5} + a\cdot 7^{6} +O\left(7^{ 7 }\right)$ $r_{ 5 }$ $=$ $5 a + 1 + \left(3 a + 2\right)\cdot 7 + \left(a + 6\right)\cdot 7^{2} + 5\cdot 7^{3} + 2 a\cdot 7^{4} + \left(a + 1\right)\cdot 7^{5} + \left(a + 4\right)\cdot 7^{6} +O\left(7^{ 7 }\right)$

### 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 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.