Properties

Label 2.6400.8t7.b.a
Dimension $2$
Group $C_8:C_2$
Conductor $6400$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_8:C_2$
Conductor: \(6400\)\(\medspace = 2^{8} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.4.327680000000.1
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: odd
Determinant: 1.5.4t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{5})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 60x^{4} - 160x^{2} + 20 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 281 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 3 + 89\cdot 281 + 89\cdot 281^{2} + 236\cdot 281^{3} + 166\cdot 281^{4} + 53\cdot 281^{5} + 94\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 73 + 126\cdot 281 + 197\cdot 281^{2} + 16\cdot 281^{3} + 110\cdot 281^{4} + 242\cdot 281^{5} + 64\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 121 + 107\cdot 281 + 164\cdot 281^{2} + 212\cdot 281^{3} + 100\cdot 281^{4} + 131\cdot 281^{5} + 20\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 134 + 108\cdot 281 + 8\cdot 281^{2} + 249\cdot 281^{3} + 13\cdot 281^{4} + 120\cdot 281^{5} + 162\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 147 + 172\cdot 281 + 272\cdot 281^{2} + 31\cdot 281^{3} + 267\cdot 281^{4} + 160\cdot 281^{5} + 118\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 160 + 173\cdot 281 + 116\cdot 281^{2} + 68\cdot 281^{3} + 180\cdot 281^{4} + 149\cdot 281^{5} + 260\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 208 + 154\cdot 281 + 83\cdot 281^{2} + 264\cdot 281^{3} + 170\cdot 281^{4} + 38\cdot 281^{5} + 216\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 278 + 191\cdot 281 + 191\cdot 281^{2} + 44\cdot 281^{3} + 114\cdot 281^{4} + 227\cdot 281^{5} + 186\cdot 281^{6} +O(281^{7})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.