Properties

Label 1.31.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 31 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$31 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 3 x^{4} - 11 x^{3} + 44 x^{2} - 36 x + 32 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{31}(26,\cdot)\)

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:
$r_{ 1 }$ $=$ $ 22 a + 10 + \left(13 a + 16\right)\cdot 23 + \left(14 a + 7\right)\cdot 23^{2} + \left(12 a + 1\right)\cdot 23^{3} + \left(9 a + 7\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + \left(8 a + 14\right)\cdot 23 + \left(16 a + 4\right)\cdot 23^{2} + \left(20 a + 16\right)\cdot 23^{3} + \left(22 a + 12\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ a + 8 + \left(9 a + 22\right)\cdot 23 + \left(8 a + 22\right)\cdot 23^{2} + \left(10 a + 11\right)\cdot 23^{3} + \left(13 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 3 a + 17 + \left(14 a + 11\right)\cdot 23 + \left(6 a + 5\right)\cdot 23^{2} + \left(2 a + 18\right)\cdot 23^{3} + 14\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 5 a + 1 + \left(a + 15\right)\cdot 23 + \left(5 a + 9\right)\cdot 23^{2} + \left(21 a + 3\right)\cdot 23^{3} + \left(6 a + 14\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 18 a + 11 + \left(21 a + 12\right)\cdot 23 + \left(17 a + 18\right)\cdot 23^{2} + \left(a + 17\right)\cdot 23^{3} + \left(16 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
$(1,2,6)(3,4,5)$
$(1,3)(2,4)(5,6)$

Character values on conjugacy classes

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