# Properties

 Label 2.53361.8t5.a.a Dimension $2$ Group $Q_8$ Conductor $53361$ Root number $-1$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$53361$$$$\medspace = 3^{2} \cdot 7^{2} \cdot 11^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin number field: Galois closure of 8.8.151939915084881.1 Galois orbit size: $1$ Smallest permutation container: $Q_8$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{21}, \sqrt{33})$$

## Defining polynomial

 $f(x)$ $=$ $x^{8} - 3 x^{7} - 62 x^{6} + 66 x^{5} + 1125 x^{4} + 264 x^{3} - 4982 x^{2} - 4245 x + 823$.

The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $6 + 140\cdot 167 + 18\cdot 167^{2} + 39\cdot 167^{3} + 151\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 2 }$ $=$ $9 + 11\cdot 167 + 160\cdot 167^{2} + 89\cdot 167^{3} + 105\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 3 }$ $=$ $22 + 119\cdot 167 + 72\cdot 167^{2} + 159\cdot 167^{3} + 117\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 4 }$ $=$ $29 + 159\cdot 167 + 119\cdot 167^{2} + 95\cdot 167^{3} + 154\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 5 }$ $=$ $31 + 4\cdot 167 + 82\cdot 167^{2} + 40\cdot 167^{3} + 164\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 6 }$ $=$ $88 + 14\cdot 167 + 73\cdot 167^{2} + 95\cdot 167^{3} + 119\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 7 }$ $=$ $154 + 152\cdot 167 + 91\cdot 167^{2} + 12\cdot 167^{3} + 84\cdot 167^{4} +O\left(167^{ 5 }\right)$ $r_{ 8 }$ $=$ $165 + 66\cdot 167 + 49\cdot 167^{2} + 135\cdot 167^{3} + 104\cdot 167^{4} +O\left(167^{ 5 }\right)$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,7)(2,3)(4,8)(5,6)$ $-2$ $2$ $4$ $(1,3,7,2)(4,6,8,5)$ $0$ $2$ $4$ $(1,8,7,4)(2,5,3,6)$ $0$ $2$ $4$ $(1,6,7,5)(2,8,3,4)$ $0$

The blue line marks the conjugacy class containing complex conjugation.