Properties

Label 2.2880.8t17.a
Dimension 2
Group $C_4\wr C_2$
Conductor $ 2^{6} \cdot 3^{2} \cdot 5 $
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:$2880= 2^{6} \cdot 3^{2} \cdot 5 $
Artin number field: Splitting field of $f= x^{8} - 6 x^{4} + 45 $ 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 $\Q_{ 41 }$ to precision 11.
Roots:
$r_{ 1 }$ $=$ $ 2 + 3\cdot 41 + 6\cdot 41^{2} + 33\cdot 41^{3} + 33\cdot 41^{4} + 20\cdot 41^{5} + 24\cdot 41^{6} + 34\cdot 41^{7} + 15\cdot 41^{8} + 26\cdot 41^{9} + 18\cdot 41^{10} +O\left(41^{ 11 }\right)$
$r_{ 2 }$ $=$ $ 12 + 33\cdot 41 + 33\cdot 41^{2} + 29\cdot 41^{3} + 19\cdot 41^{4} + 17\cdot 41^{5} + 27\cdot 41^{6} + 37\cdot 41^{7} + 10\cdot 41^{8} + 19\cdot 41^{9} + 31\cdot 41^{10} +O\left(41^{ 11 }\right)$
$r_{ 3 }$ $=$ $ 15 + 2\cdot 41 + 13\cdot 41^{2} + 40\cdot 41^{3} + 7\cdot 41^{4} + 25\cdot 41^{5} + 24\cdot 41^{6} + 8\cdot 41^{7} + 21\cdot 41^{8} + 14\cdot 41^{9} + 31\cdot 41^{10} +O\left(41^{ 11 }\right)$
$r_{ 4 }$ $=$ $ 18 + 4\cdot 41 + 27\cdot 41^{2} + 8\cdot 41^{3} + 22\cdot 41^{4} + 20\cdot 41^{5} + 34\cdot 41^{6} + 8\cdot 41^{7} + 33\cdot 41^{8} + 3\cdot 41^{9} +O\left(41^{ 11 }\right)$
$r_{ 5 }$ $=$ $ 23 + 36\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 18\cdot 41^{4} + 20\cdot 41^{5} + 6\cdot 41^{6} + 32\cdot 41^{7} + 7\cdot 41^{8} + 37\cdot 41^{9} + 40\cdot 41^{10} +O\left(41^{ 11 }\right)$
$r_{ 6 }$ $=$ $ 26 + 38\cdot 41 + 27\cdot 41^{2} + 33\cdot 41^{4} + 15\cdot 41^{5} + 16\cdot 41^{6} + 32\cdot 41^{7} + 19\cdot 41^{8} + 26\cdot 41^{9} + 9\cdot 41^{10} +O\left(41^{ 11 }\right)$
$r_{ 7 }$ $=$ $ 29 + 7\cdot 41 + 7\cdot 41^{2} + 11\cdot 41^{3} + 21\cdot 41^{4} + 23\cdot 41^{5} + 13\cdot 41^{6} + 3\cdot 41^{7} + 30\cdot 41^{8} + 21\cdot 41^{9} + 9\cdot 41^{10} +O\left(41^{ 11 }\right)$
$r_{ 8 }$ $=$ $ 39 + 37\cdot 41 + 34\cdot 41^{2} + 7\cdot 41^{3} + 7\cdot 41^{4} + 20\cdot 41^{5} + 16\cdot 41^{6} + 6\cdot 41^{7} + 25\cdot 41^{8} + 14\cdot 41^{9} + 22\cdot 41^{10} +O\left(41^{ 11 }\right)$

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

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

Character values on conjugacy classes

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