Properties

Label 2.14400.8t7.f.a
Dimension $2$
Group $C_8:C_2$
Conductor $14400$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(14400\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.4.414720000000.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: odd
Determinant: 1.40.4t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{5})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 135x^{4} - 270x^{2} + 405 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 139 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 29 + 120\cdot 139 + 138\cdot 139^{2} + 55\cdot 139^{3} + 7\cdot 139^{4} + 42\cdot 139^{5} + 84\cdot 139^{6} + 3\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 38 + 106\cdot 139 + 17\cdot 139^{2} + 44\cdot 139^{3} + 55\cdot 139^{4} + 42\cdot 139^{5} + 38\cdot 139^{6} + 58\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 41 + 7\cdot 139 + 101\cdot 139^{2} + 137\cdot 139^{3} + 136\cdot 139^{4} + 80\cdot 139^{5} + 33\cdot 139^{6} + 62\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 64 + 121\cdot 139 + 15\cdot 139^{2} + 18\cdot 139^{3} + 60\cdot 139^{4} + 13\cdot 139^{5} + 40\cdot 139^{6} + 28\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 75 + 17\cdot 139 + 123\cdot 139^{2} + 120\cdot 139^{3} + 78\cdot 139^{4} + 125\cdot 139^{5} + 98\cdot 139^{6} + 110\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 98 + 131\cdot 139 + 37\cdot 139^{2} + 139^{3} + 2\cdot 139^{4} + 58\cdot 139^{5} + 105\cdot 139^{6} + 76\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 101 + 32\cdot 139 + 121\cdot 139^{2} + 94\cdot 139^{3} + 83\cdot 139^{4} + 96\cdot 139^{5} + 100\cdot 139^{6} + 80\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 110 + 18\cdot 139 + 83\cdot 139^{3} + 131\cdot 139^{4} + 96\cdot 139^{5} + 54\cdot 139^{6} + 135\cdot 139^{7} +O(139^{8})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.