# Properties

 Label 2.2624.8t6.a Dimension $2$ Group $D_{8}$ Conductor $2624$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{8}$ Conductor: $$2624$$$$\medspace = 2^{6} \cdot 41$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 8.0.4516806656.3 Galois orbit size: $2$ Smallest permutation container: $D_{8}$ Parity: odd Projective image: $D_4$ Projective field: 4.0.656.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 8.
Roots:
 $r_{ 1 }$ $=$ $$5 + 3\cdot 61 + 23\cdot 61^{2} + 27\cdot 61^{3} + 36\cdot 61^{4} + 26\cdot 61^{5} + 53\cdot 61^{6} + 53\cdot 61^{7} +O(61^{8})$$ $r_{ 2 }$ $=$ $$20 + 17\cdot 61 + 33\cdot 61^{2} + 32\cdot 61^{3} + 29\cdot 61^{4} + 5\cdot 61^{5} + 32\cdot 61^{6} + 19\cdot 61^{7} +O(61^{8})$$ $r_{ 3 }$ $=$ $$21 + 47\cdot 61 + 36\cdot 61^{2} + 24\cdot 61^{3} + 26\cdot 61^{4} + 35\cdot 61^{5} + 7\cdot 61^{6} + 45\cdot 61^{7} +O(61^{8})$$ $r_{ 4 }$ $=$ $$22 + 30\cdot 61 + 22\cdot 61^{2} + 60\cdot 61^{3} + 54\cdot 61^{4} + 55\cdot 61^{5} + 5\cdot 61^{6} + 22\cdot 61^{7} +O(61^{8})$$ $r_{ 5 }$ $=$ $$39 + 30\cdot 61 + 38\cdot 61^{2} + 6\cdot 61^{4} + 5\cdot 61^{5} + 55\cdot 61^{6} + 38\cdot 61^{7} +O(61^{8})$$ $r_{ 6 }$ $=$ $$40 + 13\cdot 61 + 24\cdot 61^{2} + 36\cdot 61^{3} + 34\cdot 61^{4} + 25\cdot 61^{5} + 53\cdot 61^{6} + 15\cdot 61^{7} +O(61^{8})$$ $r_{ 7 }$ $=$ $$41 + 43\cdot 61 + 27\cdot 61^{2} + 28\cdot 61^{3} + 31\cdot 61^{4} + 55\cdot 61^{5} + 28\cdot 61^{6} + 41\cdot 61^{7} +O(61^{8})$$ $r_{ 8 }$ $=$ $$56 + 57\cdot 61 + 37\cdot 61^{2} + 33\cdot 61^{3} + 24\cdot 61^{4} + 34\cdot 61^{5} + 7\cdot 61^{6} + 7\cdot 61^{7} +O(61^{8})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 8 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $1$ $2$ $(1,8)(2,7)(3,6)(4,5)$ $-2$ $-2$ $4$ $2$ $(1,7)(2,8)(3,5)(4,6)$ $0$ $0$ $4$ $2$ $(2,6)(3,7)(4,5)$ $0$ $0$ $2$ $4$ $(1,5,8,4)(2,3,7,6)$ $0$ $0$ $2$ $8$ $(1,3,4,2,8,6,5,7)$ $-\zeta_{8}^{3} + \zeta_{8}$ $\zeta_{8}^{3} - \zeta_{8}$ $2$ $8$ $(1,2,5,3,8,7,4,6)$ $\zeta_{8}^{3} - \zeta_{8}$ $-\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.