Properties

Label 2.399.8t6.c.b
Dimension $2$
Group $D_{8}$
Conductor $399$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(399\)\(\medspace = 3 \cdot 7 \cdot 19 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.190563597.3
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.399.2t1.a.a
Projective image: $D_4$
Projective stem field: Galois closure of 4.0.1197.2

Defining polynomial

$f(x)$$=$ \( x^{8} - x^{7} + 2x^{6} - 6x^{5} + 17x^{4} + 3x^{3} - 7x^{2} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 12 + 41\cdot 97 + 5\cdot 97^{2} + 60\cdot 97^{3} + 20\cdot 97^{4} + 70\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 16\cdot 97 + 96\cdot 97^{2} + 29\cdot 97^{3} + 25\cdot 97^{4} + 92\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 13\cdot 97 + 15\cdot 97^{2} + 53\cdot 97^{3} + 12\cdot 97^{4} + 4\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 + 79\cdot 97^{2} + 2\cdot 97^{3} + 89\cdot 97^{4} + 74\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 67 + 48\cdot 97 + 58\cdot 97^{2} + 61\cdot 97^{3} + 39\cdot 97^{4} + 54\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 71 + 19\cdot 97 + 33\cdot 97^{2} + 50\cdot 97^{3} + 93\cdot 97^{4} + 89\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 77 + 64\cdot 97 + 80\cdot 97^{2} + 77\cdot 97^{3} + 79\cdot 97^{4} + 45\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 94 + 86\cdot 97 + 19\cdot 97^{2} + 52\cdot 97^{3} + 27\cdot 97^{4} + 53\cdot 97^{5} +O(97^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.