Properties

Label 2.3724.6t5.a
Dimension $2$
Group $S_3\times C_3$
Conductor $3724$
Indicator $0$

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Basic invariants

Dimension:$2$
Group:$S_3\times C_3$
Conductor:\(3724\)\(\medspace = 2^{2} \cdot 7^{2} \cdot 19 \)
Artin number field: Galois closure of 6.0.263495344.3
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.76.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \(x^{2} + 24 x + 2\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 5 a + 18 + \left(15 a + 23\right)\cdot 29 + 2 a\cdot 29^{2} + \left(7 a + 24\right)\cdot 29^{3} + 17 a\cdot 29^{4} + 5\cdot 29^{5} + \left(19 a + 23\right)\cdot 29^{6} +O(29^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( a + 19 + \left(24 a + 7\right)\cdot 29 + \left(9 a + 7\right)\cdot 29^{2} + \left(22 a + 14\right)\cdot 29^{3} + \left(28 a + 19\right)\cdot 29^{4} + \left(26 a + 20\right)\cdot 29^{5} + \left(16 a + 28\right)\cdot 29^{6} +O(29^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 24 a + 14 + \left(13 a + 7\right)\cdot 29 + \left(26 a + 27\right)\cdot 29^{2} + \left(21 a + 27\right)\cdot 29^{3} + \left(11 a + 21\right)\cdot 29^{4} + \left(28 a + 19\right)\cdot 29^{5} + \left(9 a + 1\right)\cdot 29^{6} +O(29^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 23 a + 8 + \left(13 a + 10\right)\cdot 29 + \left(28 a + 3\right)\cdot 29^{2} + \left(26 a + 15\right)\cdot 29^{3} + \left(16 a + 24\right)\cdot 29^{4} + \left(19 a + 3\right)\cdot 29^{5} + \left(15 a + 2\right)\cdot 29^{6} +O(29^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 28 a + 24 + \left(4 a + 10\right)\cdot 29 + \left(19 a + 3\right)\cdot 29^{2} + 6 a\cdot 29^{3} + 25\cdot 29^{4} + \left(2 a + 10\right)\cdot 29^{5} + \left(12 a + 28\right)\cdot 29^{6} +O(29^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 6 a + 7 + \left(15 a + 27\right)\cdot 29 + 15\cdot 29^{2} + \left(2 a + 5\right)\cdot 29^{3} + \left(12 a + 24\right)\cdot 29^{4} + \left(9 a + 26\right)\cdot 29^{5} + \left(13 a + 2\right)\cdot 29^{6} +O(29^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

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