Properties

Label 1.133.6t1.j.b
Dimension 1
Group $C_6$
Conductor $ 7 \cdot 19 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$133= 7 \cdot 19 $
Artin number field: Splitting field of 6.0.44700103.1 defined by $f= x^{6} - x^{5} - 7 x^{4} - 5 x^{3} + 53 x^{2} + 97 x + 121 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{133}(125,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 20 a + 23 + \left(25 a + 20\right)\cdot 31 + \left(17 a + 13\right)\cdot 31^{2} + \left(8 a + 15\right)\cdot 31^{3} + \left(27 a + 3\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 15 + \left(25 a + 12\right)\cdot 31 + \left(17 a + 13\right)\cdot 31^{2} + \left(8 a + 17\right)\cdot 31^{3} + \left(27 a + 21\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 a + 24 + \left(5 a + 12\right)\cdot 31 + \left(13 a + 23\right)\cdot 31^{2} + \left(22 a + 16\right)\cdot 31^{3} + \left(3 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 20 + \left(5 a + 28\right)\cdot 31 + \left(13 a + 29\right)\cdot 31^{2} + \left(22 a + 13\right)\cdot 31^{3} + \left(3 a + 29\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 1 + \left(5 a + 21\right)\cdot 31 + \left(13 a + 23\right)\cdot 31^{2} + \left(22 a + 14\right)\cdot 31^{3} + \left(3 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 20 a + 11 + \left(25 a + 28\right)\cdot 31 + \left(17 a + 19\right)\cdot 31^{2} + \left(8 a + 14\right)\cdot 31^{3} + \left(27 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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