Properties

Label 2.2016.8t11.c.a
Dimension $2$
Group $Q_8:C_2$
Conductor $2016$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \)
Artin stem field: Galois closure of 8.0.1792336896.7
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.56.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-6}, \sqrt{-7})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 9x^{6} + 45x^{4} - 72x^{2} + 36 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 16 + 77\cdot 79 + 54\cdot 79^{2} + 26\cdot 79^{3} + 48\cdot 79^{4} + 30\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 24 + 58\cdot 79 + 55\cdot 79^{2} + 16\cdot 79^{3} + 68\cdot 79^{4} + 69\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 37 + 17\cdot 79 + 3\cdot 79^{2} + 49\cdot 79^{3} + 44\cdot 79^{4} + 33\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 39 + 75\cdot 79 + 46\cdot 79^{2} + 46\cdot 79^{3} + 49\cdot 79^{4} + 5\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 40 + 3\cdot 79 + 32\cdot 79^{2} + 32\cdot 79^{3} + 29\cdot 79^{4} + 73\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 42 + 61\cdot 79 + 75\cdot 79^{2} + 29\cdot 79^{3} + 34\cdot 79^{4} + 45\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 55 + 20\cdot 79 + 23\cdot 79^{2} + 62\cdot 79^{3} + 10\cdot 79^{4} + 9\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 63 + 79 + 24\cdot 79^{2} + 52\cdot 79^{3} + 30\cdot 79^{4} + 48\cdot 79^{5} +O(79^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.