Properties

Label 1.3_37.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 3 \cdot 37 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$111= 3 \cdot 37 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 13 x^{4} + 34 x^{3} + 133 x^{2} + 132 x + 121 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{111}(47,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 16 a + 5 + 13\cdot 29 + \left(5 a + 4\right)\cdot 29^{2} + \left(27 a + 2\right)\cdot 29^{3} + \left(27 a + 27\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 a + 24 + \left(24 a + 24\right)\cdot 29 + 28\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + \left(27 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 27 + \left(28 a + 28\right)\cdot 29 + \left(23 a + 28\right)\cdot 29^{2} + \left(a + 16\right)\cdot 29^{3} + \left(a + 23\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 17 a + 18 + \left(7 a + 21\right)\cdot 29 + 10 a\cdot 29^{2} + \left(10 a + 2\right)\cdot 29^{3} + 15 a\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 27 + \left(4 a + 13\right)\cdot 29 + \left(28 a + 8\right)\cdot 29^{2} + 9 a\cdot 29^{3} + \left(a + 1\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 12 a + 16 + \left(21 a + 13\right)\cdot 29 + \left(18 a + 15\right)\cdot 29^{2} + \left(18 a + 14\right)\cdot 29^{3} + \left(13 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

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

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

Character values on conjugacy classes

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