Properties

Label 2.3200.8t11.e.a
Dimension $2$
Group $Q_8:C_2$
Conductor $3200$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $Q_8:C_2$
Conductor: \(3200\)\(\medspace = 2^{7} \cdot 5^{2}\)
Artin stem field: Galois closure of 8.0.4096000000.1
Galois orbit size: $2$
Smallest permutation container: $Q_8:C_2$
Parity: odd
Determinant: 1.40.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\zeta_{8})\)

Defining polynomial

$f(x)$$=$ \( x^{8} - 10x^{6} + 50x^{4} - 100x^{2} + 100 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 1 + 15\cdot 41 + 32\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} + 32\cdot 41^{5} + 2\cdot 41^{6} + 39\cdot 41^{7} +O(41^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 4\cdot 41 + 2\cdot 41^{2} + 2\cdot 41^{3} + 9\cdot 41^{4} + 33\cdot 41^{5} + 22\cdot 41^{6} + 9\cdot 41^{7} +O(41^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 14 + 25\cdot 41 + 40\cdot 41^{2} + 19\cdot 41^{3} + 30\cdot 41^{4} + 19\cdot 41^{5} + 30\cdot 41^{6} + 14\cdot 41^{7} +O(41^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 16 + 34\cdot 41 + 15\cdot 41^{3} + 20\cdot 41^{4} + 29\cdot 41^{5} + 8\cdot 41^{6} +O(41^{8})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 25 + 6\cdot 41 + 40\cdot 41^{2} + 25\cdot 41^{3} + 20\cdot 41^{4} + 11\cdot 41^{5} + 32\cdot 41^{6} + 40\cdot 41^{7} +O(41^{8})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 27 + 15\cdot 41 + 21\cdot 41^{3} + 10\cdot 41^{4} + 21\cdot 41^{5} + 10\cdot 41^{6} + 26\cdot 41^{7} +O(41^{8})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 34 + 36\cdot 41 + 38\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} + 7\cdot 41^{5} + 18\cdot 41^{6} + 31\cdot 41^{7} +O(41^{8})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 40 + 25\cdot 41 + 8\cdot 41^{2} + 38\cdot 41^{3} + 32\cdot 41^{4} + 8\cdot 41^{5} + 38\cdot 41^{6} + 41^{7} +O(41^{8})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.