Properties

 Label 1.31.6t1.a.a Dimension 1 Group $C_6$ Conductor $31$ Root number not computed Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $31$ Artin number field: Splitting field of 6.0.28629151.1 defined by $f= x^{6} - x^{5} + 3 x^{4} - 11 x^{3} + 44 x^{2} - 36 x + 32$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Corresponding Dirichlet character: $$\chi_{31}(26,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $x^{2} + 21 x + 5$
Roots:
 $r_{ 1 }$ $=$ $22 a + 10 + \left(13 a + 16\right)\cdot 23 + \left(14 a + 7\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(9 a + 7\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 a + \left(8 a + 14\right)\cdot 23 + \left(16 a + 4\right)\cdot 23^{2} + \left(20 a + 16\right)\cdot 23^{3} + \left(22 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 3 }$ $=$ $a + 8 + \left(9 a + 22\right)\cdot 23 + \left(8 a + 22\right)\cdot 23^{2} + \left(10 a + 11\right)\cdot 23^{3} + \left(13 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 4 }$ $=$ $3 a + 17 + \left(14 a + 11\right)\cdot 23 + \left(6 a + 5\right)\cdot 23^{2} + \left(2 a + 18\right)\cdot 23^{3} + 14\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 5 }$ $=$ $5 a + 1 + \left(a + 15\right)\cdot 23 + \left(5 a + 9\right)\cdot 23^{2} + \left(21 a + 3\right)\cdot 23^{3} + \left(6 a + 14\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$ $r_{ 6 }$ $=$ $18 a + 11 + \left(21 a + 12\right)\cdot 23 + \left(17 a + 18\right)\cdot 23^{2} + \left(a + 17\right)\cdot 23^{3} + \left(16 a + 6\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

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