Properties

Label 2.152.6t5.a
Dimension $2$
Group $S_3\times C_3$
Conductor $152$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:\(152\)\(\medspace = 2^{3} \cdot 19 \)
Artin number field: Galois closure of 6.0.184832.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.2888.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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