Properties

Label 2.14400.8t7.h.b
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 8.0.414720000000.3
Galois orbit size: $2$
Smallest permutation container: $C_8:C_2$
Parity: even
Determinant: 1.40.4t1.a.b
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{5})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 30x^{6} + 225x^{4} + 540x^{2} + 405 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 4 + 40\cdot 41 + 37\cdot 41^{3} + 9\cdot 41^{4} + 8\cdot 41^{5} + 4\cdot 41^{6} + 27\cdot 41^{7} + 24\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 14\cdot 41 + 12\cdot 41^{2} + 16\cdot 41^{3} + 33\cdot 41^{4} + 12\cdot 41^{5} + 38\cdot 41^{6} + 34\cdot 41^{7} + 19\cdot 41^{8} + 17\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 7 + 7\cdot 41 + 7\cdot 41^{2} + 26\cdot 41^{3} + 37\cdot 41^{4} + 39\cdot 41^{5} + 17\cdot 41^{6} + 29\cdot 41^{7} + 38\cdot 41^{8} + 17\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 8\cdot 41 + 9\cdot 41^{3} + 31\cdot 41^{4} + 3\cdot 41^{5} + 23\cdot 41^{6} + 23\cdot 41^{7} + 33\cdot 41^{8} + 33\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 + 32\cdot 41 + 40\cdot 41^{2} + 31\cdot 41^{3} + 9\cdot 41^{4} + 37\cdot 41^{5} + 17\cdot 41^{6} + 17\cdot 41^{7} + 7\cdot 41^{8} + 7\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 34 + 33\cdot 41 + 33\cdot 41^{2} + 14\cdot 41^{3} + 3\cdot 41^{4} + 41^{5} + 23\cdot 41^{6} + 11\cdot 41^{7} + 2\cdot 41^{8} + 23\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 35 + 26\cdot 41 + 28\cdot 41^{2} + 24\cdot 41^{3} + 7\cdot 41^{4} + 28\cdot 41^{5} + 2\cdot 41^{6} + 6\cdot 41^{7} + 21\cdot 41^{8} + 23\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 37 + 40\cdot 41^{2} + 3\cdot 41^{3} + 31\cdot 41^{4} + 32\cdot 41^{5} + 36\cdot 41^{6} + 13\cdot 41^{7} + 16\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.