Properties

Label 2.14400.8t7.e.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 8.8.414720000000.1
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:
$r_{ 1 }$ $=$ \( 5 + 13\cdot 41 + 27\cdot 41^{2} + 10\cdot 41^{3} + 41^{4} + 15\cdot 41^{5} + 20\cdot 41^{7} + 28\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 41 + 27\cdot 41^{2} + 9\cdot 41^{3} + 27\cdot 41^{4} + 19\cdot 41^{5} + 24\cdot 41^{6} + 36\cdot 41^{7} + 23\cdot 41^{8} + 25\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 13 + 17\cdot 41 + 32\cdot 41^{2} + 9\cdot 41^{3} + 10\cdot 41^{4} + 20\cdot 41^{5} + 35\cdot 41^{6} + 5\cdot 41^{7} + 31\cdot 41^{8} + 16\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 19 + 36\cdot 41 + 41^{2} + 40\cdot 41^{3} + 3\cdot 41^{4} + 33\cdot 41^{5} + 30\cdot 41^{6} + 5\cdot 41^{7} + 40\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 + 4\cdot 41 + 39\cdot 41^{2} + 37\cdot 41^{4} + 7\cdot 41^{5} + 10\cdot 41^{6} + 35\cdot 41^{7} + 26\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 28 + 23\cdot 41 + 8\cdot 41^{2} + 31\cdot 41^{3} + 30\cdot 41^{4} + 20\cdot 41^{5} + 5\cdot 41^{6} + 35\cdot 41^{7} + 9\cdot 41^{8} + 24\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 34 + 39\cdot 41 + 13\cdot 41^{2} + 31\cdot 41^{3} + 13\cdot 41^{4} + 21\cdot 41^{5} + 16\cdot 41^{6} + 4\cdot 41^{7} + 17\cdot 41^{8} + 15\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 36 + 27\cdot 41 + 13\cdot 41^{2} + 30\cdot 41^{3} + 39\cdot 41^{4} + 25\cdot 41^{5} + 40\cdot 41^{6} + 20\cdot 41^{7} + 12\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\) Copy content Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.