Properties

Label 2.2268.6t5.k
Dimension $2$
Group $S_3\times C_3$
Conductor $2268$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:\(2268\)\(\medspace = 2^{2} \cdot 3^{4} \cdot 7 \)
Artin number field: Galois closure of 6.0.15431472.5
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.5292.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{2} + 45x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 10 a + 22 + \left(37 a + 34\right)\cdot 47 + \left(38 a + 18\right)\cdot 47^{2} + \left(34 a + 20\right)\cdot 47^{3} + \left(32 a + 2\right)\cdot 47^{4} + \left(34 a + 15\right)\cdot 47^{5} + \left(24 a + 18\right)\cdot 47^{6} + \left(45 a + 37\right)\cdot 47^{7} +O(47^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 37 a + 42 + \left(9 a + 4\right)\cdot 47 + \left(8 a + 12\right)\cdot 47^{2} + \left(12 a + 4\right)\cdot 47^{3} + \left(14 a + 33\right)\cdot 47^{4} + \left(12 a + 4\right)\cdot 47^{5} + \left(22 a + 33\right)\cdot 47^{6} + \left(a + 9\right)\cdot 47^{7} +O(47^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 40 a + 17 + \left(5 a + 6\right)\cdot 47 + \left(12 a + 23\right)\cdot 47^{2} + \left(21 a + 22\right)\cdot 47^{3} + \left(9 a + 21\right)\cdot 47^{4} + \left(23 a + 36\right)\cdot 47^{5} + \left(40 a + 38\right)\cdot 47^{6} + \left(13 a + 20\right)\cdot 47^{7} +O(47^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 2 + \left(43 a + 17\right)\cdot 47 + \left(3 a + 40\right)\cdot 47^{2} + \left(9 a + 36\right)\cdot 47^{3} + \left(42 a + 41\right)\cdot 47^{4} + \left(10 a + 15\right)\cdot 47^{5} + \left(18 a + 11\right)\cdot 47^{6} + \left(12 a + 29\right)\cdot 47^{7} +O(47^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 44 a + 8 + \left(3 a + 6\right)\cdot 47 + \left(43 a + 5\right)\cdot 47^{2} + \left(37 a + 4\right)\cdot 47^{3} + \left(4 a + 23\right)\cdot 47^{4} + \left(36 a + 42\right)\cdot 47^{5} + \left(28 a + 36\right)\cdot 47^{6} + \left(34 a + 35\right)\cdot 47^{7} +O(47^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 a + 3 + \left(41 a + 25\right)\cdot 47 + \left(34 a + 41\right)\cdot 47^{2} + \left(25 a + 5\right)\cdot 47^{3} + \left(37 a + 19\right)\cdot 47^{4} + \left(23 a + 26\right)\cdot 47^{5} + \left(6 a + 2\right)\cdot 47^{6} + \left(33 a + 8\right)\cdot 47^{7} +O(47^{8})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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