Properties

Label 2.448.8t17.b.b
Dimension $2$
Group $C_4\wr C_2$
Conductor $448$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(448\)\(\medspace = 2^{6} \cdot 7 \)
Artin stem field: Galois closure of 8.0.78675968.2
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.112.4t1.b.b
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.14336.1

Defining polynomial

$f(x)$$=$ \( x^{8} + 3x^{6} + 3x^{4} + 2x^{2} + 2 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 127 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 13 + 18\cdot 127 + 121\cdot 127^{2} + 22\cdot 127^{3} + 60\cdot 127^{4} + 70\cdot 127^{5} + 26\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 32 + 116\cdot 127 + 49\cdot 127^{2} + 123\cdot 127^{3} + 65\cdot 127^{4} + 93\cdot 127^{5} + 91\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 34 + 110\cdot 127 + 28\cdot 127^{2} + 59\cdot 127^{3} + 111\cdot 127^{4} + 47\cdot 127^{5} + 121\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 51 + 17\cdot 127 + 8\cdot 127^{2} + 24\cdot 127^{3} + 123\cdot 127^{4} + 64\cdot 127^{5} + 83\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 76 + 109\cdot 127 + 118\cdot 127^{2} + 102\cdot 127^{3} + 3\cdot 127^{4} + 62\cdot 127^{5} + 43\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 93 + 16\cdot 127 + 98\cdot 127^{2} + 67\cdot 127^{3} + 15\cdot 127^{4} + 79\cdot 127^{5} + 5\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 95 + 10\cdot 127 + 77\cdot 127^{2} + 3\cdot 127^{3} + 61\cdot 127^{4} + 33\cdot 127^{5} + 35\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 114 + 108\cdot 127 + 5\cdot 127^{2} + 104\cdot 127^{3} + 66\cdot 127^{4} + 56\cdot 127^{5} + 100\cdot 127^{6} +O(127^{7})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.