Properties

 Label 2.278784.8t5.k.a Dimension 2 Group $Q_8$ Conductor $2^{8} \cdot 3^{2} \cdot 11^{2}$ Root number -1 Frobenius-Schur indicator -1

Related objects

Basic invariants

 Dimension: $2$ Group: $Q_8$ Conductor: $278784= 2^{8} \cdot 3^{2} \cdot 11^{2}$ Artin number field: Splitting field of 8.8.2407470785888256.1 defined by $f= x^{8} - 132 x^{6} + 2772 x^{4} - 13068 x^{2} + 9801$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $Q_8$ Parity: Even Determinant: 1.1.1t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{11})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 12.
Roots:
 $r_{ 1 }$ $=$ $5 + 26\cdot 97 + 68\cdot 97^{2} + 41\cdot 97^{3} + 95\cdot 97^{4} + 9\cdot 97^{5} + 31\cdot 97^{6} + 40\cdot 97^{7} + 62\cdot 97^{8} + 13\cdot 97^{9} + 58\cdot 97^{10} + 46\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 2 }$ $=$ $32 + 84\cdot 97 + 68\cdot 97^{2} + 2\cdot 97^{3} + 26\cdot 97^{4} + 57\cdot 97^{5} + 11\cdot 97^{6} + 30\cdot 97^{7} + 88\cdot 97^{8} + 93\cdot 97^{9} + 87\cdot 97^{10} + 50\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 3 }$ $=$ $36 + 75\cdot 97 + 97^{2} + 16\cdot 97^{3} + 66\cdot 97^{4} + 19\cdot 97^{5} + 29\cdot 97^{6} + 72\cdot 97^{7} + 88\cdot 97^{8} + 80\cdot 97^{9} + 79\cdot 97^{10} + 71\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 4 }$ $=$ $42 + 91\cdot 97 + 20\cdot 97^{2} + 43\cdot 97^{3} + 96\cdot 97^{4} + 61\cdot 97^{5} + 96\cdot 97^{6} + 80\cdot 97^{7} + 25\cdot 97^{8} + 58\cdot 97^{9} + 64\cdot 97^{10} + 37\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 5 }$ $=$ $55 + 5\cdot 97 + 76\cdot 97^{2} + 53\cdot 97^{3} + 35\cdot 97^{5} + 16\cdot 97^{7} + 71\cdot 97^{8} + 38\cdot 97^{9} + 32\cdot 97^{10} + 59\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 6 }$ $=$ $61 + 21\cdot 97 + 95\cdot 97^{2} + 80\cdot 97^{3} + 30\cdot 97^{4} + 77\cdot 97^{5} + 67\cdot 97^{6} + 24\cdot 97^{7} + 8\cdot 97^{8} + 16\cdot 97^{9} + 17\cdot 97^{10} + 25\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 7 }$ $=$ $65 + 12\cdot 97 + 28\cdot 97^{2} + 94\cdot 97^{3} + 70\cdot 97^{4} + 39\cdot 97^{5} + 85\cdot 97^{6} + 66\cdot 97^{7} + 8\cdot 97^{8} + 3\cdot 97^{9} + 9\cdot 97^{10} + 46\cdot 97^{11} +O\left(97^{ 12 }\right)$ $r_{ 8 }$ $=$ $92 + 70\cdot 97 + 28\cdot 97^{2} + 55\cdot 97^{3} + 97^{4} + 87\cdot 97^{5} + 65\cdot 97^{6} + 56\cdot 97^{7} + 34\cdot 97^{8} + 83\cdot 97^{9} + 38\cdot 97^{10} + 50\cdot 97^{11} +O\left(97^{ 12 }\right)$

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

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

Character values on conjugacy classes

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