# Properties

 Label 2.2e6_5e2.8t7.3c1 Dimension 2 Group $C_8:C_2$ Conductor $2^{6} \cdot 5^{2}$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $C_8:C_2$ Conductor: $1600= 2^{6} \cdot 5^{2}$ Artin number field: Splitting field of $f= x^{8} - 15 x^{4} + 10 x^{2} + 5$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_8:C_2$ Parity: Odd Determinant: 1.2e3_5.4t1.2c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 7.
Roots: \begin{aligned} r_{ 1 } &= 18 + 24\cdot 59 + 25\cdot 59^{2} + 3\cdot 59^{3} + 46\cdot 59^{4} + 18\cdot 59^{5} + 14\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 2 } &= 20 + 17\cdot 59 + 19\cdot 59^{2} + 6\cdot 59^{3} + 46\cdot 59^{4} + 10\cdot 59^{5} + 54\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 3 } &= 23 + 45\cdot 59 + 21\cdot 59^{2} + 59^{3} + 41\cdot 59^{4} + 4\cdot 59^{5} + 40\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 4 } &= 24 + 31\cdot 59 + 55\cdot 59^{2} + 56\cdot 59^{3} + 5\cdot 59^{4} + 44\cdot 59^{5} + 23\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 5 } &= 35 + 27\cdot 59 + 3\cdot 59^{2} + 2\cdot 59^{3} + 53\cdot 59^{4} + 14\cdot 59^{5} + 35\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 6 } &= 36 + 13\cdot 59 + 37\cdot 59^{2} + 57\cdot 59^{3} + 17\cdot 59^{4} + 54\cdot 59^{5} + 18\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 7 } &= 39 + 41\cdot 59 + 39\cdot 59^{2} + 52\cdot 59^{3} + 12\cdot 59^{4} + 48\cdot 59^{5} + 4\cdot 59^{6} +O\left(59^{ 7 }\right) \\ r_{ 8 } &= 41 + 34\cdot 59 + 33\cdot 59^{2} + 55\cdot 59^{3} + 12\cdot 59^{4} + 40\cdot 59^{5} + 44\cdot 59^{6} +O\left(59^{ 7 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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