Properties

Label 4.2e2_13_17e2.6t13.2
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{2} \cdot 13 \cdot 17^{2}$
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$15028= 2^{2} \cdot 13 \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{6} - x^{5} - 2 x^{4} + x^{3} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 15 a + 16 + \left(36 a + 2\right)\cdot 43 + \left(15 a + 28\right)\cdot 43^{2} + \left(37 a + 6\right)\cdot 43^{3} + \left(16 a + 27\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 28 a + 31 + \left(6 a + 23\right)\cdot 43 + \left(27 a + 7\right)\cdot 43^{2} + \left(5 a + 28\right)\cdot 43^{3} + \left(26 a + 6\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 30 + 41\cdot 43 + 28\cdot 43^{2} + 16\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 12 + 33\cdot 43 + 13\cdot 43^{2} + 25\cdot 43^{3} + 31\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 30 a + 27 + \left(21 a + 39\right)\cdot 43 + \left(30 a + 20\right)\cdot 43^{2} + \left(28 a + 5\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 13 a + 14 + \left(21 a + 31\right)\cdot 43 + \left(12 a + 29\right)\cdot 43^{2} + \left(14 a + 3\right)\cdot 43^{3} + \left(36 a + 25\right)\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $4$
$6$ $2$ $(1,4)(2,5)(3,6)$ $0$
$6$ $2$ $(2,3)$ $2$
$9$ $2$ $(2,3)(5,6)$ $0$
$4$ $3$ $(1,2,3)(4,5,6)$ $-2$
$4$ $3$ $(1,2,3)$ $1$
$18$ $4$ $(1,4)(2,6,3,5)$ $0$
$12$ $6$ $(1,5,2,6,3,4)$ $0$
$12$ $6$ $(2,3)(4,5,6)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.