Properties

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

Related objects

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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:
$r_{ 1 }$ $=$ \( 5 + 3\cdot 61 + 23\cdot 61^{2} + 27\cdot 61^{3} + 36\cdot 61^{4} + 26\cdot 61^{5} + 53\cdot 61^{6} + 53\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 17\cdot 61 + 33\cdot 61^{2} + 32\cdot 61^{3} + 29\cdot 61^{4} + 5\cdot 61^{5} + 32\cdot 61^{6} + 19\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 21 + 47\cdot 61 + 36\cdot 61^{2} + 24\cdot 61^{3} + 26\cdot 61^{4} + 35\cdot 61^{5} + 7\cdot 61^{6} + 45\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 22 + 30\cdot 61 + 22\cdot 61^{2} + 60\cdot 61^{3} + 54\cdot 61^{4} + 55\cdot 61^{5} + 5\cdot 61^{6} + 22\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 39 + 30\cdot 61 + 38\cdot 61^{2} + 6\cdot 61^{4} + 5\cdot 61^{5} + 55\cdot 61^{6} + 38\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 40 + 13\cdot 61 + 24\cdot 61^{2} + 36\cdot 61^{3} + 34\cdot 61^{4} + 25\cdot 61^{5} + 53\cdot 61^{6} + 15\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 41 + 43\cdot 61 + 27\cdot 61^{2} + 28\cdot 61^{3} + 31\cdot 61^{4} + 55\cdot 61^{5} + 28\cdot 61^{6} + 41\cdot 61^{7} +O(61^{8})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 56 + 57\cdot 61 + 37\cdot 61^{2} + 33\cdot 61^{3} + 24\cdot 61^{4} + 34\cdot 61^{5} + 7\cdot 61^{6} + 7\cdot 61^{7} +O(61^{8})\)  Toggle raw display

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

SizeOrderAction 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.