Properties

Label 2.2e8_3e2_7e2.8t5.4c1
Dimension 2
Group $Q_8$
Conductor $ 2^{8} \cdot 3^{2} \cdot 7^{2}$
Root number -1
Frobenius-Schur indicator -1

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$112896= 2^{8} \cdot 3^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} + 84 x^{6} + 2268 x^{4} + 19404 x^{2} + 441 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $Q_8$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 157 }$ to precision 9.
Roots:
$r_{ 1 }$ $=$ $ 34 + 17\cdot 157 + 56\cdot 157^{2} + 65\cdot 157^{3} + 131\cdot 157^{4} + 142\cdot 157^{5} + 85\cdot 157^{6} + 4\cdot 157^{7} + 35\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 2 }$ $=$ $ 37 + 41\cdot 157 + 13\cdot 157^{2} + 8\cdot 157^{3} + 31\cdot 157^{4} + 131\cdot 157^{5} + 47\cdot 157^{6} + 95\cdot 157^{7} + 97\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 3 }$ $=$ $ 67 + 2\cdot 157 + 144\cdot 157^{2} + 58\cdot 157^{3} + 55\cdot 157^{4} + 11\cdot 157^{5} + 8\cdot 157^{6} + 147\cdot 157^{7} + 30\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 4 }$ $=$ $ 76 + 123\cdot 157 + 68\cdot 157^{2} + 52\cdot 157^{3} + 118\cdot 157^{4} + 29\cdot 157^{5} + 67\cdot 157^{6} + 93\cdot 157^{7} + 138\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 5 }$ $=$ $ 81 + 33\cdot 157 + 88\cdot 157^{2} + 104\cdot 157^{3} + 38\cdot 157^{4} + 127\cdot 157^{5} + 89\cdot 157^{6} + 63\cdot 157^{7} + 18\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 6 }$ $=$ $ 90 + 154\cdot 157 + 12\cdot 157^{2} + 98\cdot 157^{3} + 101\cdot 157^{4} + 145\cdot 157^{5} + 148\cdot 157^{6} + 9\cdot 157^{7} + 126\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 7 }$ $=$ $ 120 + 115\cdot 157 + 143\cdot 157^{2} + 148\cdot 157^{3} + 125\cdot 157^{4} + 25\cdot 157^{5} + 109\cdot 157^{6} + 61\cdot 157^{7} + 59\cdot 157^{8} +O\left(157^{ 9 }\right)$
$r_{ 8 }$ $=$ $ 123 + 139\cdot 157 + 100\cdot 157^{2} + 91\cdot 157^{3} + 25\cdot 157^{4} + 14\cdot 157^{5} + 71\cdot 157^{6} + 152\cdot 157^{7} + 121\cdot 157^{8} +O\left(157^{ 9 }\right)$

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

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

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$$4$$(1,2,8,7)(3,5,6,4)$$0$
$2$$4$$(1,6,8,3)(2,5,7,4)$$0$
$2$$4$$(1,5,8,4)(2,3,7,6)$$0$
The blue line marks the conjugacy class containing complex conjugation.