Properties

Label 2.84.6t5.b.a
Dimension $2$
Group $S_3\times C_3$
Conductor $84$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Artin stem field: 6.0.21168.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.21.6t1.a.b
Projective image: $S_3$
Projective stem field: 3.1.588.1

Defining polynomial

$f(x)$$=$\(x^{6} - 3 x^{5} + 4 x^{4} - x^{3} - 2 x^{2} + x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 17 + \left(8 a + 17\right)\cdot 29 + \left(25 a + 14\right)\cdot 29^{2} + \left(24 a + 20\right)\cdot 29^{3} + \left(3 a + 3\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 a + 28 + \left(27 a + 15\right)\cdot 29 + \left(11 a + 12\right)\cdot 29^{2} + \left(20 a + 19\right)\cdot 29^{3} + \left(21 a + 22\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 8 a + 6 + \left(20 a + 11\right)\cdot 29 + \left(3 a + 16\right)\cdot 29^{2} + \left(4 a + 3\right)\cdot 29^{3} + \left(25 a + 27\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 17 a + 1 + \left(a + 25\right)\cdot 29 + \left(17 a + 15\right)\cdot 29^{2} + \left(8 a + 22\right)\cdot 29^{3} + \left(7 a + 23\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 6 a + 4 + \left(2 a + 6\right)\cdot 29 + \left(a + 12\right)\cdot 29^{2} + \left(14 a + 19\right)\cdot 29^{3} + 4 a\cdot 29^{4} +O(29^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 23 a + 5 + \left(26 a + 11\right)\cdot 29 + \left(27 a + 15\right)\cdot 29^{2} + \left(14 a + 1\right)\cdot 29^{3} + \left(24 a + 9\right)\cdot 29^{4} +O(29^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$3$$2$$(1,3)(2,4)(5,6)$$0$
$1$$3$$(1,4,5)(2,6,3)$$2 \zeta_{3}$
$1$$3$$(1,5,4)(2,3,6)$$-2 \zeta_{3} - 2$
$2$$3$$(1,5,4)$$-\zeta_{3}$
$2$$3$$(1,4,5)$$\zeta_{3} + 1$
$2$$3$$(1,5,4)(2,6,3)$$-1$
$3$$6$$(1,6,4,3,5,2)$$0$
$3$$6$$(1,2,5,3,4,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.