Properties

Label 1.3e2_7.6t1.6c2
Dimension 1
Group $C_6$
Conductor $ 3^{2} \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$63= 3^{2} \cdot 7 $
Artin number field: Splitting field of $f= x^{6} - 14 x^{3} + 63 x^{2} + 168 x + 161 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{63}(31,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 100 a + 28 + \left(35 a + 30\right)\cdot 101 + \left(35 a + 92\right)\cdot 101^{2} + \left(41 a + 55\right)\cdot 101^{3} + \left(74 a + 52\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 2 }$ $=$ $ a + 24 + \left(65 a + 74\right)\cdot 101 + \left(65 a + 96\right)\cdot 101^{2} + \left(59 a + 84\right)\cdot 101^{3} + \left(26 a + 5\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 10 a + \left(24 a + 94\right)\cdot 101 + \left(83 a + 47\right)\cdot 101^{2} + \left(56 a + 54\right)\cdot 101^{3} + \left(42 a + 43\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 91 a + 40 + \left(76 a + 79\right)\cdot 101 + \left(17 a + 53\right)\cdot 101^{2} + \left(44 a + 97\right)\cdot 101^{3} + \left(58 a + 55\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 11 a + 33 + \left(89 a + 92\right)\cdot 101 + \left(47 a + 55\right)\cdot 101^{2} + \left(15 a + 48\right)\cdot 101^{3} + \left(69 a + 93\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 90 a + 77 + \left(11 a + 33\right)\cdot 101 + \left(53 a + 57\right)\cdot 101^{2} + \left(85 a + 62\right)\cdot 101^{3} + \left(31 a + 51\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$

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

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

Character values on conjugacy classes

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