Properties

Label 1.5_13.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 5 \cdot 13 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$65= 5 \cdot 13 $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 12 x^{4} + 13 x^{3} + 19 x^{2} - 10 x - 5 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{65}(29,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 19 a + 27 + \left(21 a + 35\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(9 a + 7\right)\cdot 47^{3} + \left(30 a + 17\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 a + 43 + \left(21 a + 23\right)\cdot 47 + \left(29 a + 24\right)\cdot 47^{2} + \left(9 a + 34\right)\cdot 47^{3} + \left(30 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 28 a + 34 + 25 a\cdot 47 + \left(17 a + 15\right)\cdot 47^{2} + \left(37 a + 24\right)\cdot 47^{3} + \left(16 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 28 a + 29 + \left(25 a + 22\right)\cdot 47 + \left(17 a + 2\right)\cdot 47^{2} + \left(37 a + 10\right)\cdot 47^{3} + \left(16 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 28 a + 18 + \left(25 a + 12\right)\cdot 47 + \left(17 a + 15\right)\cdot 47^{2} + \left(37 a + 44\right)\cdot 47^{3} + \left(16 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 19 a + 38 + \left(21 a + 45\right)\cdot 47 + \left(29 a + 11\right)\cdot 47^{2} + \left(9 a + 20\right)\cdot 47^{3} + \left(30 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

Character values on conjugacy classes

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