# Properties

 Label 4.11457.6t13.b.a Dimension 4 Group $C_3^2:D_4$ Conductor $3^{2} \cdot 19 \cdot 67$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $11457= 3^{2} \cdot 19 \cdot 67$ Artin number field: Splitting field of 6.0.34371.1 defined by $f= x^{6} - x^{5} - 2 x^{3} + 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_3^2:D_4$ Parity: Even Determinant: 1.1273.2t1.a.a Projective image: $S_3\wr C_2$ Projective field: Galois closure of 6.0.34371.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $9 a + 23 + \left(15 a + 19\right)\cdot 31 + \left(21 a + 17\right)\cdot 31^{2} + \left(26 a + 14\right)\cdot 31^{3} + \left(a + 22\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $22 a + 10 + \left(15 a + 10\right)\cdot 31 + \left(9 a + 14\right)\cdot 31^{2} + \left(4 a + 15\right)\cdot 31^{3} + \left(29 a + 30\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $24 + 17\cdot 31 + 5\cdot 31^{2} + 23\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $2 a + 25 + \left(11 a + 15\right)\cdot 31 + \left(9 a + 21\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(11 a + 28\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $14 + 24\cdot 31 + 4\cdot 31^{2} + 25\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $29 a + 29 + \left(19 a + 4\right)\cdot 31 + \left(21 a + 29\right)\cdot 31^{2} + \left(5 a + 27\right)\cdot 31^{3} + \left(19 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,4)(2,5)(3,6)$ $0$ $6$ $2$ $(2,3)$ $2$ $9$ $2$ $(2,3)(5,6)$ $0$ $4$ $3$ $(1,2,3)(4,5,6)$ $-2$ $4$ $3$ $(1,2,3)$ $1$ $18$ $4$ $(1,4)(2,6,3,5)$ $0$ $12$ $6$ $(1,5,2,6,3,4)$ $0$ $12$ $6$ $(2,3)(4,5,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.