Properties

Label 2.1476.8t6.a.b
Dimension $2$
Group $D_{8}$
Conductor $1476$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(1476\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 41 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.1429145856.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.164.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.656.1

Defining polynomial

$f(x)$$=$\(x^{8} - 2 x^{7} + 5 x^{6} - 2 x^{5} + 21 x^{4} - 30 x^{3} + 75 x^{2} - 62 x + 58\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 21 + 93\cdot 197 + 77\cdot 197^{2} + 102\cdot 197^{3} + 196\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 39 + 150\cdot 197 + 15\cdot 197^{2} + 79\cdot 197^{3} + 172\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 61 + 31\cdot 197 + 175\cdot 197^{2} + 38\cdot 197^{3} + 183\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 101 + 8\cdot 197 + 162\cdot 197^{2} + 139\cdot 197^{3} + 29\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 104 + 28\cdot 197 + 20\cdot 197^{2} + 173\cdot 197^{3} + 103\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 112 + 79\cdot 197 + 82\cdot 197^{2} + 181\cdot 197^{3} + 119\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 161 + 15\cdot 197 + 72\cdot 197^{2} + 167\cdot 197^{3} + 47\cdot 197^{4} +O(197^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 191 + 183\cdot 197 + 182\cdot 197^{2} + 102\cdot 197^{3} + 131\cdot 197^{4} +O(197^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.