Properties

Label 1.7_167.6t1.2c1
Dimension 1
Group $C_6$
Conductor $ 7 \cdot 167 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1169= 7 \cdot 167 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 121 x^{4} - 81 x^{3} + 5255 x^{2} - 1681 x + 81269 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1169}(333,\cdot)\)

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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