Properties

Label 2.112896.8t5.c.a
Dimension $2$
Group $Q_8$
Conductor $112896$
Root number $-1$
Indicator $-1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $Q_8$
Conductor: \(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \)
Frobenius-Schur indicator: $-1$
Root number: $-1$
Artin field: Galois closure of 8.0.1438916737499136.1
Galois orbit size: $1$
Smallest permutation container: $Q_8$
Parity: even
Determinant: 1.1.1t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{3}, \sqrt{14})\)

Defining polynomial

$f(x)$$=$ \( x^{8} + 84x^{6} + 1260x^{4} + 5292x^{2} + 441 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 5 + 61 + 25\cdot 61^{2} + 16\cdot 61^{3} + 19\cdot 61^{4} + 55\cdot 61^{5} + 57\cdot 61^{6} + 61^{7} + 8\cdot 61^{8} + 38\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 + 23\cdot 61 + 3\cdot 61^{2} + 43\cdot 61^{3} + 2\cdot 61^{4} + 37\cdot 61^{5} + 23\cdot 61^{6} + 14\cdot 61^{7} + 39\cdot 61^{8} + 8\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 17 + 11\cdot 61 + 50\cdot 61^{2} + 11\cdot 61^{3} + 49\cdot 61^{4} + 27\cdot 61^{5} + 61^{6} + 29\cdot 61^{7} + 23\cdot 61^{8} + 48\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 23 + 40\cdot 61 + 61^{2} + 9\cdot 61^{3} + 48\cdot 61^{4} + 3\cdot 61^{5} + 35\cdot 61^{6} + 30\cdot 61^{7} + 41\cdot 61^{8} + 3\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 38 + 20\cdot 61 + 59\cdot 61^{2} + 51\cdot 61^{3} + 12\cdot 61^{4} + 57\cdot 61^{5} + 25\cdot 61^{6} + 30\cdot 61^{7} + 19\cdot 61^{8} + 57\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 44 + 49\cdot 61 + 10\cdot 61^{2} + 49\cdot 61^{3} + 11\cdot 61^{4} + 33\cdot 61^{5} + 59\cdot 61^{6} + 31\cdot 61^{7} + 37\cdot 61^{8} + 12\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 54 + 37\cdot 61 + 57\cdot 61^{2} + 17\cdot 61^{3} + 58\cdot 61^{4} + 23\cdot 61^{5} + 37\cdot 61^{6} + 46\cdot 61^{7} + 21\cdot 61^{8} + 52\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 56 + 59\cdot 61 + 35\cdot 61^{2} + 44\cdot 61^{3} + 41\cdot 61^{4} + 5\cdot 61^{5} + 3\cdot 61^{6} + 59\cdot 61^{7} + 52\cdot 61^{8} + 22\cdot 61^{9} +O(61^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character valueComplex conjugation
$1$$1$$()$$2$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-2$
$2$$4$$(1,3,8,6)(2,5,7,4)$$0$
$2$$4$$(1,4,8,5)(2,3,7,6)$$0$
$2$$4$$(1,7,8,2)(3,5,6,4)$$0$