Properties

Label 2.1400.8t6.b
Dimension $2$
Group $D_{8}$
Conductor $1400$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:\(1400\)\(\medspace = 2^{3} \cdot 5^{2} \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.0.19208000000.2
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Projective image: $D_4$
Projective field: 4.0.9800.2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 113 }$ to precision 7.
Roots:
$r_{ 1 }$ $=$ \( 24 + 78\cdot 113 + 107\cdot 113^{2} + 41\cdot 113^{3} + 70\cdot 113^{4} + 109\cdot 113^{5} + 31\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 48 + 113 + 65\cdot 113^{2} + 9\cdot 113^{3} + 89\cdot 113^{4} + 72\cdot 113^{5} + 46\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 54 + 5\cdot 113 + 98\cdot 113^{2} + 2\cdot 113^{3} + 35\cdot 113^{4} + 111\cdot 113^{5} + 47\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 56 + 52\cdot 113 + 16\cdot 113^{2} + 90\cdot 113^{3} + 12\cdot 113^{4} + 52\cdot 113^{5} + 44\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 57 + 60\cdot 113 + 96\cdot 113^{2} + 22\cdot 113^{3} + 100\cdot 113^{4} + 60\cdot 113^{5} + 68\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 59 + 107\cdot 113 + 14\cdot 113^{2} + 110\cdot 113^{3} + 77\cdot 113^{4} + 113^{5} + 65\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 65 + 111\cdot 113 + 47\cdot 113^{2} + 103\cdot 113^{3} + 23\cdot 113^{4} + 40\cdot 113^{5} + 66\cdot 113^{6} +O(113^{7})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 89 + 34\cdot 113 + 5\cdot 113^{2} + 71\cdot 113^{3} + 42\cdot 113^{4} + 3\cdot 113^{5} + 81\cdot 113^{6} +O(113^{7})\)  Toggle raw display

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

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

Character values on conjugacy classes

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