# Properties

 Label 2.112896.8t5.g.a Dimension $2$ Group $Q_8$ Conductor $112896$ Root number $-1$ Indicator $-1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $$112896$$$$\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2}$$ Frobenius-Schur indicator: $-1$ Root number: $-1$ Artin number field: Galois closure of 8.8.359729184374784.2 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 $$\Q(\sqrt{6}, \sqrt{14})$$

## Defining polynomial

 $f(x)$ $=$ $x^{8} - 84 x^{6} + 1890 x^{4} - 10584 x^{2} + 1764$.

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 11.

Roots:
 $r_{ 1 }$ $=$ $13 + 62\cdot 67 + 27\cdot 67^{2} + 51\cdot 67^{3} + 5\cdot 67^{4} + 66\cdot 67^{5} + 36\cdot 67^{6} + 40\cdot 67^{7} + 3\cdot 67^{8} + 45\cdot 67^{9} + 57\cdot 67^{10} +O\left(67^{ 11 }\right)$ $r_{ 2 }$ $=$ $16 + 9\cdot 67 + 27\cdot 67^{2} + 2\cdot 67^{3} + 44\cdot 67^{4} + 14\cdot 67^{5} + 46\cdot 67^{6} + 25\cdot 67^{7} + 40\cdot 67^{8} + 63\cdot 67^{9} +O\left(67^{ 11 }\right)$ $r_{ 3 }$ $=$ $17 + 48\cdot 67 + 66\cdot 67^{2} + 53\cdot 67^{3} + 55\cdot 67^{4} + 16\cdot 67^{5} + 49\cdot 67^{6} + 40\cdot 67^{7} + 15\cdot 67^{8} + 26\cdot 67^{9} + 37\cdot 67^{10} +O\left(67^{ 11 }\right)$ $r_{ 4 }$ $=$ $24 + 7\cdot 67 + 65\cdot 67^{2} + 14\cdot 67^{3} + 63\cdot 67^{4} + 32\cdot 67^{5} + 44\cdot 67^{6} + 56\cdot 67^{7} + 54\cdot 67^{8} + 21\cdot 67^{9} + 58\cdot 67^{10} +O\left(67^{ 11 }\right)$ $r_{ 5 }$ $=$ $43 + 59\cdot 67 + 67^{2} + 52\cdot 67^{3} + 3\cdot 67^{4} + 34\cdot 67^{5} + 22\cdot 67^{6} + 10\cdot 67^{7} + 12\cdot 67^{8} + 45\cdot 67^{9} + 8\cdot 67^{10} +O\left(67^{ 11 }\right)$ $r_{ 6 }$ $=$ $50 + 18\cdot 67 + 13\cdot 67^{3} + 11\cdot 67^{4} + 50\cdot 67^{5} + 17\cdot 67^{6} + 26\cdot 67^{7} + 51\cdot 67^{8} + 40\cdot 67^{9} + 29\cdot 67^{10} +O\left(67^{ 11 }\right)$ $r_{ 7 }$ $=$ $51 + 57\cdot 67 + 39\cdot 67^{2} + 64\cdot 67^{3} + 22\cdot 67^{4} + 52\cdot 67^{5} + 20\cdot 67^{6} + 41\cdot 67^{7} + 26\cdot 67^{8} + 3\cdot 67^{9} + 66\cdot 67^{10} +O\left(67^{ 11 }\right)$ $r_{ 8 }$ $=$ $54 + 4\cdot 67 + 39\cdot 67^{2} + 15\cdot 67^{3} + 61\cdot 67^{4} + 30\cdot 67^{6} + 26\cdot 67^{7} + 63\cdot 67^{8} + 21\cdot 67^{9} + 9\cdot 67^{10} +O\left(67^{ 11 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.