# Properties

 Label 1.117.6t1.b.a Dimension 1 Group $C_6$ Conductor $3^{2} \cdot 13$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $117= 3^{2} \cdot 13$ Artin number field: Splitting field of 6.0.562166163.2 defined by $f= x^{6} - 26 x^{3} + 2197$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{117}(113,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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