Properties

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

Related objects

Learn more about

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 stem field: 8.0.19208000000.2
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.56.2t1.b.a
Projective image: $D_4$
Projective stem field: 4.0.9800.2

Defining polynomial

$f(x)$$=$\(x^{8} - 5 x^{6} + 45 x^{4} - 75 x^{2} + 50\)  Toggle raw display.

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 value
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$4$$2$$(2,4)(3,6)(5,7)$$0$
$4$$2$$(1,2)(3,5)(4,6)(7,8)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
$2$$8$$(1,2,6,5,8,7,3,4)$$-\zeta_{8}^{3} + \zeta_{8}$
$2$$8$$(1,5,3,2,8,4,6,7)$$\zeta_{8}^{3} - \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.