Properties

Label 1.13_79.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 13 \cdot 79 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1027= 13 \cdot 79 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 62 x^{4} + 141 x^{3} + 665 x^{2} - 1882 x + 1037 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{1027}(181,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 55 a + 39 + \left(6 a + 5\right)\cdot 67 + \left(26 a + 10\right)\cdot 67^{2} + \left(23 a + 63\right)\cdot 67^{3} + \left(62 a + 50\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 29 + \left(60 a + 2\right)\cdot 67 + \left(40 a + 62\right)\cdot 67^{2} + \left(43 a + 54\right)\cdot 67^{3} + \left(4 a + 57\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 a + 9 + \left(60 a + 45\right)\cdot 67 + \left(40 a + 43\right)\cdot 67^{2} + \left(43 a + 49\right)\cdot 67^{3} + \left(4 a + 37\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 55 a + 10 + \left(6 a + 30\right)\cdot 67 + \left(26 a + 31\right)\cdot 67^{2} + \left(23 a + 54\right)\cdot 67^{3} + \left(62 a + 32\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 55 a + 57 + \left(6 a + 5\right)\cdot 67 + \left(26 a + 13\right)\cdot 67^{2} + \left(23 a + 49\right)\cdot 67^{3} + \left(62 a + 12\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 12 a + 58 + \left(60 a + 44\right)\cdot 67 + \left(40 a + 40\right)\cdot 67^{2} + \left(43 a + 63\right)\cdot 67^{3} + \left(4 a + 8\right)\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

Character values on conjugacy classes

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