# Properties

 Label 9.3e12_29e6.10t26.1c1 Dimension 9 Group $A_6$ Conductor $3^{12} \cdot 29^{6}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $9$ Group: $A_6$ Conductor: $316113500535561= 3^{12} \cdot 29^{6}$ Artin number field: Splitting field of $f= x^{6} - 3 x^{3} - 3 x + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $\PSL(2,9)$ 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 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $x^{2} + 18 x + 2$
Roots:
 $r_{ 1 }$ $=$ $2 + 17\cdot 19 + 19^{2} + 6\cdot 19^{3} + 10\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 2 }$ $=$ $15 a + 14 + \left(a + 18\right)\cdot 19 + \left(8 a + 6\right)\cdot 19^{2} + \left(16 a + 12\right)\cdot 19^{3} + \left(5 a + 2\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 3 }$ $=$ $3 a + 1 + \left(9 a + 14\right)\cdot 19 + \left(5 a + 9\right)\cdot 19^{2} + \left(5 a + 8\right)\cdot 19^{3} + \left(9 a + 10\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 4 }$ $=$ $7 + 19^{3} + 8\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 5 }$ $=$ $16 a + 4 + \left(9 a + 1\right)\cdot 19 + \left(13 a + 6\right)\cdot 19^{2} + \left(13 a + 8\right)\cdot 19^{3} + \left(9 a + 14\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$ $r_{ 6 }$ $=$ $4 a + 10 + \left(17 a + 5\right)\cdot 19 + \left(10 a + 13\right)\cdot 19^{2} + \left(2 a + 1\right)\cdot 19^{3} + \left(13 a + 11\right)\cdot 19^{4} +O\left(19^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,2,3)$ $(1,2)(3,4,5,6)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $9$ $45$ $2$ $(1,2)(3,4)$ $1$ $40$ $3$ $(1,2,3)(4,5,6)$ $0$ $40$ $3$ $(1,2,3)$ $0$ $90$ $4$ $(1,2,3,4)(5,6)$ $1$ $72$ $5$ $(1,2,3,4,5)$ $-1$ $72$ $5$ $(1,3,4,5,2)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.