Properties

Label 1.2e3_3_43.6t1.1c2
Dimension 1
Group $C_6$
Conductor $ 2^{3} \cdot 3 \cdot 43 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$1032= 2^{3} \cdot 3 \cdot 43 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} - 45 x^{4} + 36 x^{3} + 308 x^{2} - 304 x - 188 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{1032}(995,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 33 a + 28 + \left(13 a + 1\right)\cdot 41 + \left(30 a + 29\right)\cdot 41^{2} + \left(15 a + 4\right)\cdot 41^{3} + \left(27 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 a + 6 + \left(27 a + 35\right)\cdot 41 + \left(10 a + 24\right)\cdot 41^{2} + \left(25 a + 32\right)\cdot 41^{3} + \left(13 a + 28\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 8 a + 4 + \left(27 a + 10\right)\cdot 41 + \left(10 a + 24\right)\cdot 41^{2} + \left(25 a + 21\right)\cdot 41^{3} + \left(13 a + 6\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 33 a + 20 + \left(13 a + 20\right)\cdot 41 + \left(30 a + 30\right)\cdot 41^{2} + \left(15 a + 15\right)\cdot 41^{3} + \left(27 a + 18\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 37 + \left(27 a + 28\right)\cdot 41 + \left(10 a + 25\right)\cdot 41^{2} + \left(25 a + 32\right)\cdot 41^{3} + \left(13 a + 2\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 33 a + 30 + \left(13 a + 26\right)\cdot 41 + \left(30 a + 29\right)\cdot 41^{2} + \left(15 a + 15\right)\cdot 41^{3} + \left(27 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

Character values on conjugacy classes

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