Properties

Label 2.57600.8t5.f
Dimension $2$
Group $Q_8$
Conductor $57600$
Indicator $-1$

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:\(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2}\)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin number field: Galois closure of 8.0.7644119040000.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{3})\)

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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