Properties

Label 2.84.6t5.b
Dimension 2
Group $S_3\times C_3$
Conductor $ 2^{2} \cdot 3 \cdot 7 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:$84= 2^{2} \cdot 3 \cdot 7 $
Artin number field: Splitting field of $f= x^{6} - 3 x^{5} + 4 x^{4} - x^{3} - 2 x^{2} + x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $S_3\times C_3$
Parity: Odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.588.1

Galois action

Roots of defining polynomial

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 $
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\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 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\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 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\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 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\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 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\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 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\left(29^{ 5 }\right)$

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 values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$3$ $2$ $(1,3)(2,4)(5,6)$ $0$ $0$
$1$ $3$ $(1,4,5)(2,6,3)$ $2 \zeta_{3}$ $-2 \zeta_{3} - 2$
$1$ $3$ $(1,5,4)(2,3,6)$ $-2 \zeta_{3} - 2$ $2 \zeta_{3}$
$2$ $3$ $(1,5,4)$ $-\zeta_{3}$ $\zeta_{3} + 1$
$2$ $3$ $(1,4,5)$ $\zeta_{3} + 1$ $-\zeta_{3}$
$2$ $3$ $(1,5,4)(2,6,3)$ $-1$ $-1$
$3$ $6$ $(1,6,4,3,5,2)$ $0$ $0$
$3$ $6$ $(1,2,5,3,4,6)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.