Properties

 Label 1.171.6t1.e Dimension $1$ Group $C_6$ Conductor $171$ Indicator $0$

Related objects

Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin number field: Galois closure of 6.6.48737056617.2 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even 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 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12 x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$a + 9 + \left(3 a + 6\right)\cdot 13 + \left(5 a + 6\right)\cdot 13^{2} + \left(3 a + 12\right)\cdot 13^{3} + 6\cdot 13^{4} + \left(a + 6\right)\cdot 13^{5} + \left(a + 7\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 2 }$ $=$ $$2 a + 3 + \left(5 a + 6\right)\cdot 13 + 11\cdot 13^{2} + \left(9 a + 9\right)\cdot 13^{3} + 3\cdot 13^{4} + \left(12 a + 5\right)\cdot 13^{5} + 4\cdot 13^{6} +O(13^{7})$$ $r_{ 3 }$ $=$ $$12 a + \left(10 a + 11\right)\cdot 13 + \left(4 a + 5\right)\cdot 13^{2} + \left(7 a + 5\right)\cdot 13^{3} + \left(12 a + 5\right)\cdot 13^{4} + a\cdot 13^{5} + 13^{6} +O(13^{7})$$ $r_{ 4 }$ $=$ $$11 a + 5 + \left(7 a + 9\right)\cdot 13 + \left(12 a + 6\right)\cdot 13^{2} + \left(3 a + 5\right)\cdot 13^{3} + \left(12 a + 8\right)\cdot 13^{4} + 3\cdot 13^{5} + \left(12 a + 6\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 5 }$ $=$ $$a + 12 + \left(2 a + 9\right)\cdot 13 + \left(8 a + 12\right)\cdot 13^{2} + \left(5 a + 7\right)\cdot 13^{3} + 10\cdot 13^{4} + \left(11 a + 2\right)\cdot 13^{5} + \left(12 a + 12\right)\cdot 13^{6} +O(13^{7})$$ $r_{ 6 }$ $=$ $$12 a + 10 + \left(9 a + 8\right)\cdot 13 + \left(7 a + 8\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(12 a + 3\right)\cdot 13^{4} + \left(11 a + 7\right)\cdot 13^{5} + \left(11 a + 7\right)\cdot 13^{6} +O(13^{7})$$

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,6)(2,4)(3,5)$ $-1$ $-1$ $1$ $3$ $(1,4,5)(2,3,6)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,5,4)(2,6,3)$ $-\zeta_{3} - 1$ $\zeta_{3}$ $1$ $6$ $(1,3,4,6,5,2)$ $\zeta_{3} + 1$ $-\zeta_{3}$ $1$ $6$ $(1,2,5,6,4,3)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.