Properties

Label 2.1792.8t8.a.b
Dimension $2$
Group $QD_{16}$
Conductor $1792$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $QD_{16}$
Conductor: \(1792\)\(\medspace = 2^{8} \cdot 7 \)
Artin stem field: 8.2.5754585088.2
Galois orbit size: $2$
Smallest permutation container: $QD_{16}$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: D_4
Projective stem field: 4.0.1568.1

Defining polynomial

$f(x)$$=$\(x^{8} + 4 x^{6} + 4 x^{4} - 4 x^{2} - 7\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 2 + 9\cdot 79 + 42\cdot 79^{2} + 18\cdot 79^{3} + 36\cdot 79^{4} + 14\cdot 79^{5} + 55\cdot 79^{6} + 47\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 77\cdot 79 + 44\cdot 79^{2} + 47\cdot 79^{3} + 27\cdot 79^{4} + 55\cdot 79^{5} + 57\cdot 79^{6} + 25\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 4\cdot 79 + 10\cdot 79^{2} + 5\cdot 79^{3} + 73\cdot 79^{4} + 42\cdot 79^{5} + 7\cdot 79^{6} + 13\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 34 + 73\cdot 79 + 11\cdot 79^{2} + 6\cdot 79^{3} + 24\cdot 79^{4} + 8\cdot 79^{5} + 17\cdot 79^{6} + 62\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 45 + 5\cdot 79 + 67\cdot 79^{2} + 72\cdot 79^{3} + 54\cdot 79^{4} + 70\cdot 79^{5} + 61\cdot 79^{6} + 16\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 71 + 74\cdot 79 + 68\cdot 79^{2} + 73\cdot 79^{3} + 5\cdot 79^{4} + 36\cdot 79^{5} + 71\cdot 79^{6} + 65\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 73 + 79 + 34\cdot 79^{2} + 31\cdot 79^{3} + 51\cdot 79^{4} + 23\cdot 79^{5} + 21\cdot 79^{6} + 53\cdot 79^{7} +O(79^{8})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 77 + 69\cdot 79 + 36\cdot 79^{2} + 60\cdot 79^{3} + 42\cdot 79^{4} + 64\cdot 79^{5} + 23\cdot 79^{6} + 31\cdot 79^{7} +O(79^{8})\)  Toggle raw display

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

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

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$$2$$(2,5)(3,6)(4,7)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
$4$$4$$(1,7,8,2)(3,4,6,5)$$0$
$2$$8$$(1,4,3,7,8,5,6,2)$$\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.