Properties

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

Related objects

Learn more

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: 8.0.19208000000.1
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)$$=$\(x^{8} - x^{7} + 9 x^{6} - 27 x^{5} + 44 x^{4} - 94 x^{3} + 156 x^{2} - 88 x + 16\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 26 + 125\cdot 193 + 123\cdot 193^{2} + 186\cdot 193^{3} + 120\cdot 193^{4} + 38\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 37 + 146\cdot 193 + 183\cdot 193^{2} + 85\cdot 193^{3} + 36\cdot 193^{4} + 61\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 49 + 51\cdot 193 + 157\cdot 193^{2} + 15\cdot 193^{3} + 90\cdot 193^{4} + 6\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 51 + 61\cdot 193 + 67\cdot 193^{2} + 133\cdot 193^{3} + 5\cdot 193^{4} +O(193^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 59 + 82\cdot 193 + 180\cdot 193^{2} + 114\cdot 193^{3} + 92\cdot 193^{4} + 92\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 86 + 29\cdot 193 + 154\cdot 193^{2} + 175\cdot 193^{3} + 77\cdot 193^{4} + 77\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 130 + 136\cdot 193 + 88\cdot 193^{2} + 83\cdot 193^{3} + 57\cdot 193^{4} + 36\cdot 193^{5} +O(193^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 142 + 139\cdot 193 + 9\cdot 193^{2} + 169\cdot 193^{3} + 97\cdot 193^{4} + 73\cdot 193^{5} +O(193^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,5)(2,3)(4,7)(6,8)$$-2$
$4$$2$$(1,7)(2,8)(3,6)(4,5)$$0$
$4$$2$$(1,6)(4,7)(5,8)$$0$
$2$$4$$(1,6,5,8)(2,4,3,7)$$0$
$2$$8$$(1,4,8,2,5,7,6,3)$$\zeta_{8}^{3} - \zeta_{8}$
$2$$8$$(1,2,6,4,5,3,8,7)$$-\zeta_{8}^{3} + \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.