Properties

 Label 1.104.6t1.d Dimension 1 Group $C_6$ Conductor $2^{3} \cdot 13$ Frobenius-Schur indicator 0

Related objects

Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $104= 2^{3} \cdot 13$ Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - x^{4} - 2 x^{3} + 34 x^{2} + 28 x + 73$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Odd Projective image: $C_1$ Projective field: $$\Q$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $x^{2} + 29 x + 3$
Roots:
 $r_{ 1 }$ $=$ $a + 3 + \left(23 a + 30\right)\cdot 31 + \left(12 a + 29\right)\cdot 31^{2} + \left(23 a + 23\right)\cdot 31^{3} + \left(6 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $30 a + 23 + \left(7 a + 26\right)\cdot 31 + \left(18 a + 25\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(24 a + 8\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $30 a + 7 + \left(7 a + 12\right)\cdot 31 + \left(18 a + 7\right)\cdot 31^{2} + \left(7 a + 16\right)\cdot 31^{3} + \left(24 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $a + 5 + \left(23 a + 29\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(23 a + 13\right)\cdot 31^{3} + \left(6 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 5 }$ $=$ $a + 21 + \left(23 a + 12\right)\cdot 31 + \left(12 a + 23\right)\cdot 31^{2} + \left(23 a + 4\right)\cdot 31^{3} + \left(6 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 6 }$ $=$ $30 a + 5 + \left(7 a + 13\right)\cdot 31 + \left(18 a + 1\right)\cdot 31^{2} + \left(7 a + 27\right)\cdot 31^{3} + \left(24 a + 15\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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