Properties

Label 1.3_37.6t1.2c2
Dimension 1
Group $C_6$
Conductor $ 3 \cdot 37 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$111= 3 \cdot 37 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 22 x^{4} - 83 x^{3} + 52 x^{2} + 36 x + 184 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{111}(11,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $ x^{2} + 7 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 5 a + 3 + \left(6 a + 7\right)\cdot 11 + \left(2 a + 9\right)\cdot 11^{2} + \left(5 a + 8\right)\cdot 11^{3} + \left(6 a + 8\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 5 + \left(6 a + 10\right)\cdot 11 + \left(10 a + 3\right)\cdot 11^{2} + \left(7 a + 5\right)\cdot 11^{3} + \left(2 a + 3\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 6 a + 1 + \left(4 a + 6\right)\cdot 11 + \left(8 a + 2\right)\cdot 11^{2} + \left(5 a + 5\right)\cdot 11^{3} + \left(4 a + 7\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 a + 5 + \left(a + 4\right)\cdot 11 + \left(7 a + 2\right)\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} +O\left(11^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 6 + \left(4 a + 10\right)\cdot 11 + 6\cdot 11^{2} + \left(3 a + 4\right)\cdot 11^{3} + \left(8 a + 6\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 6 a + 3 + \left(9 a + 5\right)\cdot 11 + \left(3 a + 7\right)\cdot 11^{2} + \left(2 a + 1\right)\cdot 11^{3} + \left(10 a + 6\right)\cdot 11^{4} +O\left(11^{ 5 }\right)$

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

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

Character values on conjugacy classes

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