Properties

Label 4.5e2_421.6t13.1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 5^{2} \cdot 421 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$10525= 5^{2} \cdot 421 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{4} - x^{3} + x^{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 $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 14 a + 7 + \left(11 a + 15\right)\cdot 31 + \left(12 a + 28\right)\cdot 31^{2} + \left(17 a + 30\right)\cdot 31^{3} + \left(22 a + 13\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 a + 16 + \left(22 a + 30\right)\cdot 31 + \left(21 a + 5\right)\cdot 31^{2} + \left(a + 24\right)\cdot 31^{3} + \left(13 a + 25\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 16 + 25\cdot 31 + 28\cdot 31^{2} + 17\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 20 + 22\cdot 31 + 22\cdot 31^{2} + 8\cdot 31^{3} + 6\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 24 a + 30 + \left(8 a + 5\right)\cdot 31 + \left(9 a + 27\right)\cdot 31^{2} + \left(29 a + 5\right)\cdot 31^{3} + \left(17 a + 19\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 17 a + 4 + \left(19 a + 24\right)\cdot 31 + \left(18 a + 10\right)\cdot 31^{2} + \left(13 a + 22\right)\cdot 31^{3} + \left(8 a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

Character values on conjugacy classes

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