# Properties

 Label 2.2e7.4t3.3c1 Dimension 2 Group $D_4$ Conductor $2^{7}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_4$ Conductor: $128= 2^{7}$ Artin number field: Splitting field of $f= x^{8} + 6 x^{4} + 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.2e3.2t1.2c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots: \begin{aligned} r_{ 1 } &= 11 + 19\cdot 41 + 9\cdot 41^{2} + 18\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 2 } &= 12 + 32\cdot 41 + 15\cdot 41^{2} + 29\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 3 } &= 15 + 11\cdot 41 + 20\cdot 41^{2} + 35\cdot 41^{3} + 35\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 4 } &= 17 + 26\cdot 41 + 17\cdot 41^{2} + 39\cdot 41^{3} +O\left(41^{ 5 }\right) \\ r_{ 5 } &= 24 + 14\cdot 41 + 23\cdot 41^{2} + 41^{3} + 40\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 6 } &= 26 + 29\cdot 41 + 20\cdot 41^{2} + 5\cdot 41^{3} + 5\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 7 } &= 29 + 8\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right) \\ r_{ 8 } &= 30 + 21\cdot 41 + 31\cdot 41^{2} + 22\cdot 41^{3} + 25\cdot 41^{4} +O\left(41^{ 5 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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