Properties

Label 2.72.6t5.b
Dimension 2
Group $S_3\times C_3$
Conductor $ 2^{3} \cdot 3^{2}$
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:$72= 2^{3} \cdot 3^{2} $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 2 x^{2} + 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 3.1.648.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 24 a + 9 + \left(21 a + 32\right)\cdot 37 + \left(4 a + 18\right)\cdot 37^{2} + \left(31 a + 8\right)\cdot 37^{3} + \left(35 a + 15\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 a + 31 + \left(15 a + 20\right)\cdot 37 + \left(32 a + 15\right)\cdot 37^{2} + \left(5 a + 17\right)\cdot 37^{3} + \left(a + 16\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 29 a + 18 + \left(35 a + 13\right)\cdot 37 + \left(27 a + 30\right)\cdot 37^{2} + \left(12 a + 29\right)\cdot 37^{3} + 35\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 26 a + 1 + \left(16 a + 12\right)\cdot 37 + \left(12 a + 27\right)\cdot 37^{2} + \left(8 a + 27\right)\cdot 37^{3} + \left(24 a + 20\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 23 + \left(a + 16\right)\cdot 37 + \left(9 a + 32\right)\cdot 37^{2} + \left(24 a + 15\right)\cdot 37^{3} + \left(36 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 11 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(24 a + 23\right)\cdot 37^{2} + \left(28 a + 11\right)\cdot 37^{3} + \left(12 a + 35\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$3$ $2$ $(1,2)(3,5)(4,6)$ $0$ $0$
$1$ $3$ $(1,6,5)(2,4,3)$ $2 \zeta_{3}$ $-2 \zeta_{3} - 2$
$1$ $3$ $(1,5,6)(2,3,4)$ $-2 \zeta_{3} - 2$ $2 \zeta_{3}$
$2$ $3$ $(1,5,6)$ $-\zeta_{3}$ $\zeta_{3} + 1$
$2$ $3$ $(1,6,5)$ $\zeta_{3} + 1$ $-\zeta_{3}$
$2$ $3$ $(1,5,6)(2,4,3)$ $-1$ $-1$
$3$ $6$ $(1,3,6,2,5,4)$ $0$ $0$
$3$ $6$ $(1,4,5,2,6,3)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.