Properties

Label 1.13.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$13 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{13}(10,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 4\cdot 31 + \left(11 a + 23\right)\cdot 31^{2} + 4\cdot 31^{3} + \left(4 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 a + 6 + \left(30 a + 1\right)\cdot 31 + \left(19 a + 14\right)\cdot 31^{2} + \left(30 a + 25\right)\cdot 31^{3} + \left(26 a + 17\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 24 a + 9 + \left(2 a + 12\right)\cdot 31 + \left(14 a + 18\right)\cdot 31^{2} + \left(27 a + 15\right)\cdot 31^{3} + 15 a\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 18 a + 24 + \left(15 a + 18\right)\cdot 31 + \left(22 a + 28\right)\cdot 31^{2} + \left(23 a + 13\right)\cdot 31^{3} + \left(20 a + 5\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 13 a + 29 + 15 a\cdot 31 + \left(8 a + 27\right)\cdot 31^{2} + \left(7 a + 7\right)\cdot 31^{3} + \left(10 a + 23\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 26 + \left(28 a + 24\right)\cdot 31 + \left(16 a + 12\right)\cdot 31^{2} + \left(3 a + 25\right)\cdot 31^{3} + \left(15 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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