# Properties

 Label 4.5e2_11e3_29.8t35.4c1 Dimension 4 Group $C_2 \wr C_2\wr C_2$ Conductor $5^{2} \cdot 11^{3} \cdot 29$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_2 \wr C_2\wr C_2$ Conductor: $964975= 5^{2} \cdot 11^{3} \cdot 29$ Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $C_2 \wr C_2\wr C_2$ Parity: Odd Determinant: 1.11_29.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 6.
Roots:
 $r_{ 1 }$ $=$ $87 + 443\cdot 929 + 885\cdot 929^{2} + 341\cdot 929^{3} + 239\cdot 929^{4} + 712\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 2 }$ $=$ $97 + 391\cdot 929 + 161\cdot 929^{2} + 803\cdot 929^{3} + 490\cdot 929^{4} + 815\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 3 }$ $=$ $233 + 383\cdot 929 + 561\cdot 929^{2} + 307\cdot 929^{3} + 682\cdot 929^{4} + 669\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 4 }$ $=$ $299 + 474\cdot 929 + 74\cdot 929^{2} + 212\cdot 929^{3} + 372\cdot 929^{4} + 757\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 5 }$ $=$ $311 + 557\cdot 929 + 336\cdot 929^{2} + 67\cdot 929^{3} + 564\cdot 929^{4} + 647\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 6 }$ $=$ $358 + 377\cdot 929 + 73\cdot 929^{2} + 545\cdot 929^{3} + 634\cdot 929^{4} + 524\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 7 }$ $=$ $680 + 821\cdot 929 + 251\cdot 929^{2} + 697\cdot 929^{3} + 420\cdot 929^{4} + 482\cdot 929^{5} +O\left(929^{ 6 }\right)$ $r_{ 8 }$ $=$ $724 + 267\cdot 929 + 442\cdot 929^{2} + 741\cdot 929^{3} + 311\cdot 929^{4} + 35\cdot 929^{5} +O\left(929^{ 6 }\right)$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character value $1$ $1$ $()$ $4$ $1$ $2$ $(1,4)(2,7)(3,5)(6,8)$ $-4$ $2$ $2$ $(2,7)(6,8)$ $0$ $4$ $2$ $(2,7)$ $2$ $4$ $2$ $(1,3)(2,6)(4,5)(7,8)$ $0$ $4$ $2$ $(2,7)(3,5)$ $0$ $4$ $2$ $(2,6)(7,8)$ $-2$ $4$ $2$ $(1,4)(2,7)(6,8)$ $-2$ $4$ $2$ $(1,3)(2,7)(4,5)(6,8)$ $2$ $8$ $2$ $(1,6)(2,3)(4,8)(5,7)$ $0$ $8$ $2$ $(1,4)(2,6)(7,8)$ $0$ $4$ $4$ $(1,3,4,5)(2,8,7,6)$ $0$ $4$ $4$ $(2,8,7,6)$ $-2$ $4$ $4$ $(1,5,4,3)(2,7)(6,8)$ $2$ $8$ $4$ $(1,6,4,8)(2,5,7,3)$ $0$ $8$ $4$ $(1,4)(2,8,7,6)$ $0$ $8$ $4$ $(1,3)(2,8,7,6)(4,5)$ $0$ $16$ $4$ $(1,6,3,2)(4,8,5,7)$ $0$ $16$ $4$ $(1,6)(2,5,7,3)(4,8)$ $0$ $16$ $8$ $(1,6,3,2,4,8,5,7)$ $0$
The blue line marks the conjugacy class containing complex conjugation.