Properties

Label 2.57.6t5.a
Dimension 2
Group $S_3\times C_3$
Conductor $ 3 \cdot 19 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:$57= 3 \cdot 19 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + x^{4} - 2 x^{3} + 4 x^{2} - 3 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 3.1.1083.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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