# Properties

 Label 2.2e6_3e2_5e2.8t7.8c1 Dimension 2 Group $C_8:C_2$ Conductor $2^{6} \cdot 3^{2} \cdot 5^{2}$ Root number not computed Frobenius-Schur indicator 0

# Learn more about

## Basic invariants

 Dimension: $2$ Group: $C_8:C_2$ Conductor: $14400= 2^{6} \cdot 3^{2} \cdot 5^{2}$ Artin number field: Splitting field of $f= x^{8} + 30 x^{6} + 240 x^{4} + 720 x^{2} + 720$ over $\Q$ Size of Galois orbit: 2 Smallest containing permutation representation: $C_8:C_2$ Parity: Even Determinant: 1.2e2_5.4t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 10.
Roots: \begin{aligned} r_{ 1 } &= 1 + 11\cdot 59 + 54\cdot 59^{2} + 4\cdot 59^{3} + 46\cdot 59^{4} + 36\cdot 59^{5} + 22\cdot 59^{6} + 8\cdot 59^{7} + 39\cdot 59^{8} + 49\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 2 } &= 13 + 7\cdot 59 + 42\cdot 59^{2} + 22\cdot 59^{3} + 3\cdot 59^{4} + 44\cdot 59^{5} + 11\cdot 59^{6} + 24\cdot 59^{7} + 11\cdot 59^{8} + 28\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 3 } &= 14 + 10\cdot 59^{2} + 51\cdot 59^{3} + 27\cdot 59^{4} + 15\cdot 59^{5} + 56\cdot 59^{6} + 34\cdot 59^{7} + 40\cdot 59^{8} + 3\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 4 } &= 28 + 55\cdot 59 + 43\cdot 59^{2} + 7\cdot 59^{3} + 8\cdot 59^{4} + 26\cdot 59^{5} + 55\cdot 59^{6} + 27\cdot 59^{7} + 45\cdot 59^{8} + 30\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 5 } &= 31 + 3\cdot 59 + 15\cdot 59^{2} + 51\cdot 59^{3} + 50\cdot 59^{4} + 32\cdot 59^{5} + 3\cdot 59^{6} + 31\cdot 59^{7} + 13\cdot 59^{8} + 28\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 6 } &= 45 + 58\cdot 59 + 48\cdot 59^{2} + 7\cdot 59^{3} + 31\cdot 59^{4} + 43\cdot 59^{5} + 2\cdot 59^{6} + 24\cdot 59^{7} + 18\cdot 59^{8} + 55\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 7 } &= 46 + 51\cdot 59 + 16\cdot 59^{2} + 36\cdot 59^{3} + 55\cdot 59^{4} + 14\cdot 59^{5} + 47\cdot 59^{6} + 34\cdot 59^{7} + 47\cdot 59^{8} + 30\cdot 59^{9} +O\left(59^{ 10 }\right) \\ r_{ 8 } &= 58 + 47\cdot 59 + 4\cdot 59^{2} + 54\cdot 59^{3} + 12\cdot 59^{4} + 22\cdot 59^{5} + 36\cdot 59^{6} + 50\cdot 59^{7} + 19\cdot 59^{8} + 9\cdot 59^{9} +O\left(59^{ 10 }\right) \\ \end{aligned}

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

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

### Character values on conjugacy classes

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