Properties

Label 1.3_31.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 3 \cdot 31 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$93= 3 \cdot 31 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 11 x^{4} - 6 x^{3} + 108 x^{2} - 80 x + 64 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{93}(56,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 2 a + 4 + \left(16 a + 10\right)\cdot 23 + \left(17 a + 4\right)\cdot 23^{2} + \left(12 a + 18\right)\cdot 23^{3} + \left(13 a + 14\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ a + 19 + \left(11 a + 13\right)\cdot 23 + \left(12 a + 9\right)\cdot 23^{2} + \left(14 a + 20\right)\cdot 23^{3} + \left(5 a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 a + 12 + \left(8 a + 20\right)\cdot 23 + \left(21 a + 9\right)\cdot 23^{2} + \left(2 a + 14\right)\cdot 23^{3} + \left(6 a + 5\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 21 + \left(11 a + 11\right)\cdot 23 + 10 a\cdot 23^{2} + \left(8 a + 14\right)\cdot 23^{3} + 17 a\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 3 a + 6 + \left(14 a + 18\right)\cdot 23 + \left(a + 20\right)\cdot 23^{2} + \left(20 a + 21\right)\cdot 23^{3} + \left(16 a + 14\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 21 a + 8 + \left(6 a + 17\right)\cdot 23 + 5 a\cdot 23^{2} + \left(10 a + 3\right)\cdot 23^{3} + \left(9 a + 6\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

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$$3$$(1,4,5)(2,3,6)$$-\zeta_{3} - 1$
$1$$3$$(1,5,4)(2,6,3)$$\zeta_{3}$
$1$$6$$(1,3,4,6,5,2)$$-\zeta_{3}$
$1$$6$$(1,2,5,6,4,3)$$\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.