# Properties

 Label 1.77.6t1.a.b Dimension 1 Group $C_6$ Conductor $7 \cdot 11$ Root number not computed Frobenius-Schur indicator 0

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $77= 7 \cdot 11$ Artin number field: Splitting field of 6.0.3195731.1 defined by $f= x^{6} - x^{5} + 4 x^{4} - 3 x^{3} + 29 x^{2} - 4 x + 71$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{77}(65,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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

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

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

### Character values on conjugacy classes

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