Properties

Label 2.1280.8t17.d.b
Dimension $2$
Group $C_4\wr C_2$
Conductor $1280$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(1280\)\(\medspace = 2^{8} \cdot 5 \)
Artin stem field: Galois closure of 8.0.2097152000.8
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.5.4t1.a.b
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.8000.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 9.

Roots:
$r_{ 1 }$ $=$ \( 5 + 31\cdot 61 + 54\cdot 61^{2} + 4\cdot 61^{3} + 49\cdot 61^{4} + 54\cdot 61^{6} + 55\cdot 61^{7} + 25\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 30\cdot 61 + 43\cdot 61^{2} + 7\cdot 61^{3} + 42\cdot 61^{4} + 6\cdot 61^{5} + 2\cdot 61^{6} + 25\cdot 61^{7} + 24\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 25 + 25\cdot 61 + 48\cdot 61^{2} + 34\cdot 61^{3} + 29\cdot 61^{4} + 24\cdot 61^{5} + 43\cdot 61^{6} + 29\cdot 61^{7} + 40\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 30 + 55\cdot 61 + 25\cdot 61^{2} + 21\cdot 61^{3} + 45\cdot 61^{4} + 20\cdot 61^{5} + 16\cdot 61^{6} + 57\cdot 61^{7} + 60\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 31 + 5\cdot 61 + 35\cdot 61^{2} + 39\cdot 61^{3} + 15\cdot 61^{4} + 40\cdot 61^{5} + 44\cdot 61^{6} + 3\cdot 61^{7} +O(61^{9})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 36 + 35\cdot 61 + 12\cdot 61^{2} + 26\cdot 61^{3} + 31\cdot 61^{4} + 36\cdot 61^{5} + 17\cdot 61^{6} + 31\cdot 61^{7} + 20\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 55 + 30\cdot 61 + 17\cdot 61^{2} + 53\cdot 61^{3} + 18\cdot 61^{4} + 54\cdot 61^{5} + 58\cdot 61^{6} + 35\cdot 61^{7} + 36\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 56 + 29\cdot 61 + 6\cdot 61^{2} + 56\cdot 61^{3} + 11\cdot 61^{4} + 60\cdot 61^{5} + 6\cdot 61^{6} + 5\cdot 61^{7} + 35\cdot 61^{8} +O(61^{9})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.