Properties

Label 2.799.8t6.a.a
Dimension $2$
Group $D_{8}$
Conductor $799$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(799\)\(\medspace = 17 \cdot 47 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.2.8671400783.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.799.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.2.13583.1

Defining polynomial

$f(x)$$=$\(x^{8} - x^{7} + 2 x^{6} + x^{5} - 15 x^{4} + 23 x^{3} + 6 x^{2} - 27 x + 19\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 491 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 65 + 330\cdot 491 + 432\cdot 491^{2} + 148\cdot 491^{3} + 252\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 134 + 171\cdot 491 + 138\cdot 491^{2} + 15\cdot 491^{3} + 241\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 230 + 433\cdot 491 + 74\cdot 491^{2} + 144\cdot 491^{3} + 429\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 247 + 218\cdot 491 + 154\cdot 491^{2} + 442\cdot 491^{3} + 446\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 251 + 349\cdot 491 + 462\cdot 491^{2} + 138\cdot 491^{3} + 36\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 288 + 393\cdot 491 + 95\cdot 491^{2} + 381\cdot 491^{3} + 103\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 319 + 473\cdot 491 + 396\cdot 491^{2} + 275\cdot 491^{3} + 441\cdot 491^{4} +O(491^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 431 + 84\cdot 491 + 208\cdot 491^{2} + 417\cdot 491^{3} + 12\cdot 491^{4} +O(491^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.