# Properties

 Label 1.63.6t1.d.b Dimension 1 Group $C_6$ Conductor $3^{2} \cdot 7$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $63= 3^{2} \cdot 7$ Artin number field: Splitting field of 6.0.110270727.1 defined by $f= x^{6} - 14 x^{3} + 63 x^{2} + 168 x + 161$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{63}(31,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $x^{2} + 97 x + 2$
Roots:
 $r_{ 1 }$ $=$ $100 a + 28 + \left(35 a + 30\right)\cdot 101 + \left(35 a + 92\right)\cdot 101^{2} + \left(41 a + 55\right)\cdot 101^{3} + \left(74 a + 52\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 2 }$ $=$ $a + 24 + \left(65 a + 74\right)\cdot 101 + \left(65 a + 96\right)\cdot 101^{2} + \left(59 a + 84\right)\cdot 101^{3} + \left(26 a + 5\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 3 }$ $=$ $10 a + \left(24 a + 94\right)\cdot 101 + \left(83 a + 47\right)\cdot 101^{2} + \left(56 a + 54\right)\cdot 101^{3} + \left(42 a + 43\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 4 }$ $=$ $91 a + 40 + \left(76 a + 79\right)\cdot 101 + \left(17 a + 53\right)\cdot 101^{2} + \left(44 a + 97\right)\cdot 101^{3} + \left(58 a + 55\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 5 }$ $=$ $11 a + 33 + \left(89 a + 92\right)\cdot 101 + \left(47 a + 55\right)\cdot 101^{2} + \left(15 a + 48\right)\cdot 101^{3} + \left(69 a + 93\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ $r_{ 6 }$ $=$ $90 a + 77 + \left(11 a + 33\right)\cdot 101 + \left(53 a + 57\right)\cdot 101^{2} + \left(85 a + 62\right)\cdot 101^{3} + \left(31 a + 51\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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