# Properties

 Label 2.52.6t5.b.b Dimension 2 Group $S_3\times C_3$ Conductor $2^{2} \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $52= 2^{2} \cdot 13$ Artin number field: Splitting field of $f= x^{6} - x^{4} - 2 x^{3} + 2 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $S_3\times C_3$ Parity: Odd Determinant: 1.52.6t1.b.b

## 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:
 $r_{ 1 }$ $=$ $9 a + 19 + \left(14 a + 8\right)\cdot 31 + \left(17 a + 2\right)\cdot 31^{2} + \left(25 a + 4\right)\cdot 31^{3} + \left(23 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $6 a + 21 + \left(23 a + 13\right)\cdot 31 + \left(18 a + 21\right)\cdot 31^{2} + \left(26 a + 15\right)\cdot 31^{3} + \left(25 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $25 a + 2 + \left(7 a + 23\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(4 a + 19\right)\cdot 31^{3} + \left(5 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 27 + \left(9 a + 5\right)\cdot 31 + \left(14 a + 11\right)\cdot 31^{2} + \left(20 a + 10\right)\cdot 31^{3} + \left(28 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $22 a + 6 + \left(16 a + 28\right)\cdot 31 + \left(13 a + 22\right)\cdot 31^{2} + \left(5 a + 6\right)\cdot 31^{3} + \left(7 a + 16\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $20 a + 18 + \left(21 a + 13\right)\cdot 31 + \left(16 a + 30\right)\cdot 31^{2} + \left(10 a + 5\right)\cdot 31^{3} + \left(2 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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