Properties

 Label 1.117.6t1.f.b Dimension 1 Group $C_6$ Conductor $3^{2} \cdot 13$ Root number not computed Frobenius-Schur indicator 0

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Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $117= 3^{2} \cdot 13$ Artin number field: Splitting field of 6.6.2436053373.1 defined by $f= x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 234 x - 468$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_6$ Parity: Even Corresponding Dirichlet character: $$\chi_{117}(4,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $x^{2} + 58 x + 2$
Roots:
 $r_{ 1 }$ $=$ $3 a + 40 + \left(24 a + 2\right)\cdot 59 + \left(22 a + 30\right)\cdot 59^{2} + \left(18 a + 54\right)\cdot 59^{3} + \left(17 a + 18\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 2 }$ $=$ $45 a + 9 + \left(26 a + 55\right)\cdot 59 + \left(58 a + 57\right)\cdot 59^{2} + \left(21 a + 43\right)\cdot 59^{3} + \left(33 a + 15\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 a + 10 + \left(8 a + 1\right)\cdot 59 + \left(37 a + 30\right)\cdot 59^{2} + \left(18 a + 19\right)\cdot 59^{3} + \left(8 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 4 }$ $=$ $48 a + 21 + \left(50 a + 57\right)\cdot 59 + \left(21 a + 58\right)\cdot 59^{2} + 40 a\cdot 59^{3} + \left(50 a + 14\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 5 }$ $=$ $56 a + 43 + \left(34 a + 23\right)\cdot 59 + \left(36 a + 28\right)\cdot 59^{2} + \left(40 a + 50\right)\cdot 59^{3} + \left(41 a + 17\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ $r_{ 6 }$ $=$ $14 a + 54 + \left(32 a + 36\right)\cdot 59 + 30\cdot 59^{2} + \left(37 a + 7\right)\cdot 59^{3} + \left(25 a + 27\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

Character values on conjugacy classes

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