# Properties

 Label 4.508805.8t29.c.a Dimension 4 Group $(((C_4 \times C_2): C_2):C_2):C_2$ Conductor $5 \cdot 11^{2} \cdot 29^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $(((C_4 \times C_2): C_2):C_2):C_2$ Conductor: $508805= 5 \cdot 11^{2} \cdot 29^{2}$ Artin number field: Splitting field of 8.0.307827025.1 defined by $f= x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $(((C_4 \times C_2): C_2):C_2):C_2$ Parity: Even Determinant: 1.5.2t1.a.a Projective image: $C_2^2\wr C_2$ Projective field: Galois closure of 8.4.63600625.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 7.
Roots:
 $r_{ 1 }$ $=$ $31 + 345\cdot 929 + 280\cdot 929^{2} + 51\cdot 929^{3} + 355\cdot 929^{4} + 762\cdot 929^{5} + 249\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 2 }$ $=$ $140 + 502\cdot 929 + 7\cdot 929^{2} + 62\cdot 929^{3} + 107\cdot 929^{4} + 18\cdot 929^{5} + 279\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 3 }$ $=$ $151 + 749\cdot 929 + 698\cdot 929^{2} + 11\cdot 929^{3} + 466\cdot 929^{4} + 220\cdot 929^{5} + 41\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 4 }$ $=$ $591 + 857\cdot 929 + 639\cdot 929^{2} + 212\cdot 929^{3} + 823\cdot 929^{4} + 223\cdot 929^{5} + 250\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 5 }$ $=$ $595 + 595\cdot 929 + 818\cdot 929^{2} + 210\cdot 929^{3} + 452\cdot 929^{4} + 756\cdot 929^{5} + 702\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 6 }$ $=$ $676 + 630\cdot 929 + 522\cdot 929^{2} + 62\cdot 929^{3} + 287\cdot 929^{4} + 283\cdot 929^{5} + 254\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 7 }$ $=$ $740 + 158\cdot 929 + 686\cdot 929^{2} + 102\cdot 929^{3} + 767\cdot 929^{4} + 370\cdot 929^{5} + 441\cdot 929^{6} +O\left(929^{ 7 }\right)$ $r_{ 8 }$ $=$ $793 + 805\cdot 929 + 61\cdot 929^{2} + 215\cdot 929^{3} + 458\cdot 929^{4} + 151\cdot 929^{5} + 568\cdot 929^{6} +O\left(929^{ 7 }\right)$

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

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

### Character values on conjugacy classes

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