# Properties

 Label 2.133.6t5.a.b Dimension 2 Group $S_3\times C_3$ Conductor $7 \cdot 19$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $133= 7 \cdot 19$ Artin number field: Splitting field of 6.0.336091.1 defined by $f= x^{6} - x^{5} + x^{4} + 5 x^{2} + 4 x + 1$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $S_3\times C_3$ Parity: Odd Determinant: 1.133.6t1.i.a Projective image: $S_3$ Projective field: Galois closure of 3.1.931.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots:
 $r_{ 1 }$ $=$ $11 a + 9 + \left(5 a + 7\right)\cdot 13 + \left(10 a + 3\right)\cdot 13^{2} + \left(8 a + 10\right)\cdot 13^{3} + \left(12 a + 6\right)\cdot 13^{4} + \left(12 a + 3\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 2 }$ $=$ $4 a + 11 + \left(3 a + 10\right)\cdot 13 + \left(12 a + 3\right)\cdot 13^{2} + \left(a + 2\right)\cdot 13^{3} + 13^{4} + \left(10 a + 12\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 3 }$ $=$ $2 a + 7 + \left(7 a + 2\right)\cdot 13 + \left(2 a + 8\right)\cdot 13^{2} + \left(4 a + 8\right)\cdot 13^{3} + 10\cdot 13^{4} + 3\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 4 }$ $=$ $6 a + 9 + \left(5 a + 10\right)\cdot 13 + \left(10 a + 2\right)\cdot 13^{2} + \left(8 a + 7\right)\cdot 13^{3} + \left(11 a + 2\right)\cdot 13^{4} + \left(3 a + 9\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 5 }$ $=$ $9 a + 2 + \left(9 a + 10\right)\cdot 13 + 12\cdot 13^{2} + \left(11 a + 4\right)\cdot 13^{3} + \left(12 a + 12\right)\cdot 13^{4} + \left(2 a + 8\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$ $r_{ 6 }$ $=$ $7 a + 2 + \left(7 a + 10\right)\cdot 13 + \left(2 a + 7\right)\cdot 13^{2} + \left(4 a + 5\right)\cdot 13^{3} + \left(a + 5\right)\cdot 13^{4} + \left(9 a + 1\right)\cdot 13^{5} +O\left(13^{ 6 }\right)$

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

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

### Character values on conjugacy classes

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