Properties

Label 2.3e2_13.8t17.1
Dimension 2
Group $C_4\wr C_2$
Conductor $ 3^{2} \cdot 13 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:$117= 3^{2} \cdot 13 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{6} - 3 x^{5} + 3 x^{4} + 3 x^{3} - 2 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4\wr C_2$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O\left(139^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

Cycle notation
$(1,6,2,8)(3,5,4,7)$
$(3,7,4,5)$
$(3,4)(5,7)$
$(1,5)(2,7)(3,8)(4,6)$
$(1,2)(3,4)(5,7)(6,8)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$ $c2$
$1$ $1$ $()$ $2$ $2$
$1$ $2$ $(1,2)(3,4)(5,7)(6,8)$ $-2$ $-2$
$2$ $2$ $(3,4)(5,7)$ $0$ $0$
$4$ $2$ $(1,5)(2,7)(3,8)(4,6)$ $0$ $0$
$1$ $4$ $(1,6,2,8)(3,5,4,7)$ $-2 \zeta_{4}$ $2 \zeta_{4}$
$1$ $4$ $(1,8,2,6)(3,7,4,5)$ $2 \zeta_{4}$ $-2 \zeta_{4}$
$2$ $4$ $(3,7,4,5)$ $\zeta_{4} + 1$ $-\zeta_{4} + 1$
$2$ $4$ $(3,5,4,7)$ $-\zeta_{4} + 1$ $\zeta_{4} + 1$
$2$ $4$ $(1,2)(3,5,4,7)(6,8)$ $-\zeta_{4} - 1$ $\zeta_{4} - 1$
$2$ $4$ $(1,2)(3,7,4,5)(6,8)$ $\zeta_{4} - 1$ $-\zeta_{4} - 1$
$2$ $4$ $(1,6,2,8)(3,7,4,5)$ $0$ $0$
$4$ $4$ $(1,7,2,5)(3,8,4,6)$ $0$ $0$
$4$ $8$ $(1,3,8,7,2,4,6,5)$ $0$ $0$
$4$ $8$ $(1,7,6,3,2,5,8,4)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.