# Properties

 Label 1.3_5_7.6t1.2c1 Dimension 1 Group $C_6$ Conductor $3 \cdot 5 \cdot 7$ Root number not computed Frobenius-Schur indicator 0

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## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $105= 3 \cdot 5 \cdot 7$ Artin number field: Splitting field of $f= x^{6} - x^{5} + 7 x^{4} - 5 x^{3} + 49 x^{2} - 9 x + 139$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{105}(74,\cdot)$$

## 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:
 $r_{ 1 }$ $=$ $2 a + \left(2 a + 6\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + 9 a\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 2 }$ $=$ $11 a + 2 + \left(10 a + 6\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + 3 a\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 3 }$ $=$ $2 a + 3 + \left(2 a + 4\right)\cdot 13 + \left(10 a + 10\right)\cdot 13^{2} + \left(9 a + 7\right)\cdot 13^{3} + 11 a\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 4 }$ $=$ $11 a + 5 + \left(10 a + 4\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(3 a + 7\right)\cdot 13^{3} + \left(a + 2\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 5 }$ $=$ $2 a + 1 + \left(2 a + 9\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + \left(9 a + 4\right)\cdot 13^{3} + \left(11 a + 12\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$ $r_{ 6 }$ $=$ $11 a + 3 + \left(10 a + 9\right)\cdot 13 + \left(2 a + 6\right)\cdot 13^{2} + \left(3 a + 4\right)\cdot 13^{3} + \left(a + 1\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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