Properties

Label 2.2475.8t8.a.a
Dimension $2$
Group $QD_{16}$
Conductor $2475$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $QD_{16}$
Conductor: \(2475\)\(\medspace = 3^{2} \cdot 5^{2} \cdot 11 \)
Artin stem field: 8.2.15160921875.1
Galois orbit size: $2$
Smallest permutation container: $QD_{16}$
Parity: odd
Determinant: 1.11.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.2.2475.1

Defining polynomial

$f(x)$$=$\(x^{8} - 3 x^{7} + 7 x^{6} - 39 x^{4} + 105 x^{3} - 137 x^{2} + 96 x - 29\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 22 + 34\cdot 89 + 41\cdot 89^{2} + 48\cdot 89^{3} + 13\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 26 + 15\cdot 89 + 70\cdot 89^{2} + 32\cdot 89^{3} + 60\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 38 + 55\cdot 89 + 15\cdot 89^{2} + 35\cdot 89^{3} + 70\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 43 + 58\cdot 89 + 46\cdot 89^{2} + 69\cdot 89^{3} + 85\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 47 + 71\cdot 89 + 65\cdot 89^{2} + 17\cdot 89^{3} + 76\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 49 + 64\cdot 89 + 82\cdot 89^{2} + 65\cdot 89^{3} + 34\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 55 + 41\cdot 89^{2} + 75\cdot 89^{3} + 88\cdot 89^{4} +O(89^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 79 + 55\cdot 89 + 81\cdot 89^{2} + 10\cdot 89^{3} + 15\cdot 89^{4} +O(89^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.