Properties

Label 2.735.8t6.c.a
Dimension $2$
Group $D_{8}$
Conductor $735$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(735\)\(\medspace = 3 \cdot 5 \cdot 7^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.2.8338372875.1
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.15.2t1.a.a
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.15435.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 3x^{7} + 7x^{6} - 21x^{4} + 63x^{3} - 77x^{2} + 60x - 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 9 + 44\cdot 167 + 100\cdot 167^{2} + 50\cdot 167^{3} + 38\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 34 + 52\cdot 167 + 13\cdot 167^{2} + 51\cdot 167^{3} + 129\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 53 + 151\cdot 167 + 38\cdot 167^{2} + 21\cdot 167^{3} + 65\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 56 + 52\cdot 167 + 123\cdot 167^{2} + 90\cdot 167^{3} + 156\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 101 + 64\cdot 167 + 147\cdot 167^{2} + 122\cdot 167^{3} + 99\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 131 + 141\cdot 167 + 7\cdot 167^{2} + 50\cdot 167^{3} + 94\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 137 + 9\cdot 167 + 32\cdot 167^{2} + 109\cdot 167^{3} + 78\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 150 + 151\cdot 167 + 37\cdot 167^{2} + 5\cdot 167^{3} + 6\cdot 167^{4} +O(167^{5})\) Copy content Toggle raw display

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

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

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$$2$$(1,4)(2,8)(3,6)(5,7)$$0$
$4$$2$$(1,8)(2,7)(4,6)$$0$
$2$$4$$(1,3,8,5)(2,6,4,7)$$0$
$2$$8$$(1,6,3,4,8,7,5,2)$$-\zeta_{8}^{3} + \zeta_{8}$
$2$$8$$(1,4,5,6,8,2,3,7)$$\zeta_{8}^{3} - \zeta_{8}$

The blue line marks the conjugacy class containing complex conjugation.