Properties

Label 1.2e6.16t1.1
Dimension 1
Group $C_{16}$
Conductor $ 2^{6}$
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_{16}$
Conductor:$64= 2^{6} $
Artin number field: Splitting field of $f= x^{16} + 16 x^{14} + 104 x^{12} + 352 x^{10} + 660 x^{8} + 672 x^{6} + 336 x^{4} + 64 x^{2} + 2 $ over $\Q$
Size of Galois orbit: 8
Smallest containing permutation representation: $C_{16}$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ $ 3 + 20\cdot 31 + 4\cdot 31^{2} + 4\cdot 31^{3} + 19\cdot 31^{4} + 21\cdot 31^{5} + 15\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 5 + 12\cdot 31 + 19\cdot 31^{2} + 8\cdot 31^{3} + 12\cdot 31^{4} + 19\cdot 31^{5} + 29\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 7 + 9\cdot 31 + 31^{2} + 22\cdot 31^{3} + 25\cdot 31^{4} + 3\cdot 31^{5} + 9\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 8 + 2\cdot 31 + 27\cdot 31^{2} + 26\cdot 31^{3} + 18\cdot 31^{4} + 16\cdot 31^{5} + 16\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 9 + 4\cdot 31 + 11\cdot 31^{2} + 23\cdot 31^{4} + 17\cdot 31^{5} + 15\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 10 + 30\cdot 31 + 24\cdot 31^{2} + 30\cdot 31^{3} + 20\cdot 31^{4} + 11\cdot 31^{5} + 4\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 12 + 16\cdot 31 + 8\cdot 31^{2} + 18\cdot 31^{4} + 28\cdot 31^{5} + 2\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 15 + 20\cdot 31 + 16\cdot 31^{2} + 20\cdot 31^{3} + 23\cdot 31^{4} + 17\cdot 31^{5} + 23\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 9 }$ $=$ $ 16 + 10\cdot 31 + 14\cdot 31^{2} + 10\cdot 31^{3} + 7\cdot 31^{4} + 13\cdot 31^{5} + 7\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 10 }$ $=$ $ 19 + 14\cdot 31 + 22\cdot 31^{2} + 30\cdot 31^{3} + 12\cdot 31^{4} + 2\cdot 31^{5} + 28\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 11 }$ $=$ $ 21 + 6\cdot 31^{2} + 10\cdot 31^{4} + 19\cdot 31^{5} + 26\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 12 }$ $=$ $ 22 + 26\cdot 31 + 19\cdot 31^{2} + 30\cdot 31^{3} + 7\cdot 31^{4} + 13\cdot 31^{5} + 15\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 13 }$ $=$ $ 23 + 28\cdot 31 + 3\cdot 31^{2} + 4\cdot 31^{3} + 12\cdot 31^{4} + 14\cdot 31^{5} + 14\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 14 }$ $=$ $ 24 + 21\cdot 31 + 29\cdot 31^{2} + 8\cdot 31^{3} + 5\cdot 31^{4} + 27\cdot 31^{5} + 21\cdot 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 15 }$ $=$ $ 26 + 18\cdot 31 + 11\cdot 31^{2} + 22\cdot 31^{3} + 18\cdot 31^{4} + 11\cdot 31^{5} + 31^{6} +O\left(31^{ 7 }\right)$
$r_{ 16 }$ $=$ $ 28 + 10\cdot 31 + 26\cdot 31^{2} + 26\cdot 31^{3} + 11\cdot 31^{4} + 9\cdot 31^{5} + 15\cdot 31^{6} +O\left(31^{ 7 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 16 }$

Cycle notation
$(1,14,16,3)(2,4,15,13)(5,8,12,9)(6,10,11,7)$
$(1,2,9,11,14,4,5,7,16,15,8,6,3,13,12,10)$
$(1,8,14,12,16,9,3,5)(2,6,4,10,15,11,13,7)$
$(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 16 }$ Character values
$c1$ $c2$ $c3$ $c4$ $c5$ $c6$ $c7$ $c8$
$1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $2$ $(1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$ $-1$
$1$ $4$ $(1,14,16,3)(2,4,15,13)(5,8,12,9)(6,10,11,7)$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$
$1$ $4$ $(1,3,16,14)(2,13,15,4)(5,9,12,8)(6,7,11,10)$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$ $-\zeta_{16}^{4}$ $\zeta_{16}^{4}$
$1$ $8$ $(1,9,14,5,16,8,3,12)(2,11,4,7,15,6,13,10)$ $\zeta_{16}^{2}$ $\zeta_{16}^{6}$ $-\zeta_{16}^{2}$ $-\zeta_{16}^{6}$ $\zeta_{16}^{2}$ $\zeta_{16}^{6}$ $-\zeta_{16}^{2}$ $-\zeta_{16}^{6}$
$1$ $8$ $(1,5,3,9,16,12,14,8)(2,7,13,11,15,10,4,6)$ $\zeta_{16}^{6}$ $\zeta_{16}^{2}$ $-\zeta_{16}^{6}$ $-\zeta_{16}^{2}$ $\zeta_{16}^{6}$ $\zeta_{16}^{2}$ $-\zeta_{16}^{6}$ $-\zeta_{16}^{2}$
$1$ $8$ $(1,8,14,12,16,9,3,5)(2,6,4,10,15,11,13,7)$ $-\zeta_{16}^{2}$ $-\zeta_{16}^{6}$ $\zeta_{16}^{2}$ $\zeta_{16}^{6}$ $-\zeta_{16}^{2}$ $-\zeta_{16}^{6}$ $\zeta_{16}^{2}$ $\zeta_{16}^{6}$
$1$ $8$ $(1,12,3,8,16,5,14,9)(2,10,13,6,15,7,4,11)$ $-\zeta_{16}^{6}$ $-\zeta_{16}^{2}$ $\zeta_{16}^{6}$ $\zeta_{16}^{2}$ $-\zeta_{16}^{6}$ $-\zeta_{16}^{2}$ $\zeta_{16}^{6}$ $\zeta_{16}^{2}$
$1$ $16$ $(1,2,9,11,14,4,5,7,16,15,8,6,3,13,12,10)$ $\zeta_{16}$ $\zeta_{16}^{3}$ $\zeta_{16}^{5}$ $\zeta_{16}^{7}$ $-\zeta_{16}$ $-\zeta_{16}^{3}$ $-\zeta_{16}^{5}$ $-\zeta_{16}^{7}$
$1$ $16$ $(1,11,5,15,3,10,9,4,16,6,12,2,14,7,8,13)$ $\zeta_{16}^{3}$ $-\zeta_{16}$ $-\zeta_{16}^{7}$ $\zeta_{16}^{5}$ $-\zeta_{16}^{3}$ $\zeta_{16}$ $\zeta_{16}^{7}$ $-\zeta_{16}^{5}$
$1$ $16$ $(1,4,8,10,14,15,12,11,16,13,9,7,3,2,5,6)$ $\zeta_{16}^{5}$ $-\zeta_{16}^{7}$ $-\zeta_{16}$ $\zeta_{16}^{3}$ $-\zeta_{16}^{5}$ $\zeta_{16}^{7}$ $\zeta_{16}$ $-\zeta_{16}^{3}$
$1$ $16$ $(1,7,12,4,3,11,8,2,16,10,5,13,14,6,9,15)$ $\zeta_{16}^{7}$ $\zeta_{16}^{5}$ $\zeta_{16}^{3}$ $\zeta_{16}$ $-\zeta_{16}^{7}$ $-\zeta_{16}^{5}$ $-\zeta_{16}^{3}$ $-\zeta_{16}$
$1$ $16$ $(1,15,9,6,14,13,5,10,16,2,8,11,3,4,12,7)$ $-\zeta_{16}$ $-\zeta_{16}^{3}$ $-\zeta_{16}^{5}$ $-\zeta_{16}^{7}$ $\zeta_{16}$ $\zeta_{16}^{3}$ $\zeta_{16}^{5}$ $\zeta_{16}^{7}$
$1$ $16$ $(1,6,5,2,3,7,9,13,16,11,12,15,14,10,8,4)$ $-\zeta_{16}^{3}$ $\zeta_{16}$ $\zeta_{16}^{7}$ $-\zeta_{16}^{5}$ $\zeta_{16}^{3}$ $-\zeta_{16}$ $-\zeta_{16}^{7}$ $\zeta_{16}^{5}$
$1$ $16$ $(1,13,8,7,14,2,12,6,16,4,9,10,3,15,5,11)$ $-\zeta_{16}^{5}$ $\zeta_{16}^{7}$ $\zeta_{16}$ $-\zeta_{16}^{3}$ $\zeta_{16}^{5}$ $-\zeta_{16}^{7}$ $-\zeta_{16}$ $\zeta_{16}^{3}$
$1$ $16$ $(1,10,12,13,3,6,8,15,16,7,5,4,14,11,9,2)$ $-\zeta_{16}^{7}$ $-\zeta_{16}^{5}$ $-\zeta_{16}^{3}$ $-\zeta_{16}$ $\zeta_{16}^{7}$ $\zeta_{16}^{5}$ $\zeta_{16}^{3}$ $\zeta_{16}$
The blue line marks the conjugacy class containing complex conjugation.