Properties

Label 4.7e2_317.6t13.1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 7^{2} \cdot 317 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$15533= 7^{2} \cdot 317 $
Artin number field: Splitting field of $f= x^{6} - x^{5} + 2 x^{4} - 2 x^{3} + x^{2} + 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 $\Q_{ 23 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $ x^{2} + 21 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 18 a + 15 + \left(8 a + 18\right)\cdot 23 + \left(a + 8\right)\cdot 23^{2} + \left(16 a + 12\right)\cdot 23^{3} + \left(a + 3\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 + 18\cdot 23 + 2\cdot 23^{2} + 8\cdot 23^{3} + 11\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 3 }$ $=$ $ a + 4 + \left(11 a + 14\right)\cdot 23 + 20\cdot 23^{2} + \left(3 a + 5\right)\cdot 23^{3} + 10 a\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 a + 5 + \left(14 a + 18\right)\cdot 23 + \left(21 a + 2\right)\cdot 23^{2} + \left(6 a + 20\right)\cdot 23^{3} + \left(21 a + 13\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 22 a + 6 + \left(11 a + 12\right)\cdot 23 + \left(22 a + 10\right)\cdot 23^{2} + \left(19 a + 11\right)\cdot 23^{3} + \left(12 a + 17\right)\cdot 23^{4} +O\left(23^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 13 + 10\cdot 23 + 11\cdot 23^{3} + 22\cdot 23^{4} +O\left(23^{ 5 }\right)$

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

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

Character values on conjugacy classes

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