Properties

Label 2.1764.8t6.a.a
Dimension $2$
Group $D_{8}$
Conductor $1764$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.38423222208.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: D_4
Projective stem field: 4.0.12348.1

Defining polynomial

$f(x)$$=$\(x^{8} - 3 x^{7} + 7 x^{6} - 21 x^{5} + 42 x^{4} - 42 x^{3} + 28 x^{2} - 24 x + 16\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 233 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 13 + 205\cdot 233 + 123\cdot 233^{2} + 161\cdot 233^{3} + 158\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 36 + 167\cdot 233 + 231\cdot 233^{2} + 53\cdot 233^{3} + 94\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 83 + 16\cdot 233 + 61\cdot 233^{2} + 58\cdot 233^{3} + 80\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 115 + 89\cdot 233 + 174\cdot 233^{2} + 158\cdot 233^{3} + 211\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 146 + 193\cdot 233 + 110\cdot 233^{2} + 110\cdot 233^{3} + 185\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 154 + 101\cdot 233 + 8\cdot 233^{2} + 169\cdot 233^{3} + 19\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 183 + 73\cdot 233^{2} + 138\cdot 233^{3} + 58\cdot 233^{4} +O(233^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 205 + 157\cdot 233 + 148\cdot 233^{2} + 81\cdot 233^{3} + 123\cdot 233^{4} +O(233^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.