Properties

Label 2.1764.8t11.a.a
Dimension $2$
Group $Q_8:C_2$
Conductor $1764$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(1764\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7^{2}\)
Artin stem field: 8.0.1372257936.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{3}, \sqrt{-7})\)

Defining polynomial

$f(x)$$=$\(x^{8} - 2 x^{7} - 7 x^{6} + 7 x^{5} + 28 x^{4} - 7 x^{3} - 7 x^{2} - 79 x + 67\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 6 + 101\cdot 109 + 100\cdot 109^{2} + 45\cdot 109^{3} + 23\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 8 + 100\cdot 109 + 46\cdot 109^{2} + 30\cdot 109^{3} + 34\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 22 + 92\cdot 109 + 35\cdot 109^{2} + 70\cdot 109^{3} + 96\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 24 + 73\cdot 109 + 67\cdot 109^{2} + 86\cdot 109^{3} + 39\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 43 + 45\cdot 109 + 8\cdot 109^{2} + 26\cdot 109^{3} + 106\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 56 + 61\cdot 109 + 67\cdot 109^{2} + 30\cdot 109^{3} + 47\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 74 + 33\cdot 109 + 34\cdot 109^{2} + 71\cdot 109^{3} + 63\cdot 109^{4} +O(109^{5})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 96 + 37\cdot 109 + 74\cdot 109^{2} + 74\cdot 109^{3} + 24\cdot 109^{4} +O(109^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.