Properties

Label 1.3e2_5_23.6t1.1c1
Dimension 1
Group $C_6$
Conductor $ 3^{2} \cdot 5 \cdot 23 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1035= 3^{2} \cdot 5 \cdot 23 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 84 x^{4} - 161 x^{3} + 2607 x^{2} - 2706 x + 29699 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{1035}(229,\cdot)\)

Galois action

Roots of defining polynomial

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