Properties

Label 2.5e2_89e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 5^{2} \cdot 89^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$198025= 5^{2} \cdot 89^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 78 x^{6} - 100 x^{5} + 2541 x^{4} - 7670 x^{3} + 65772 x^{2} - 36712 x + 1485296 $ 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 $\Q_{ 139 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 21 + 81\cdot 139 + 91\cdot 139^{2} + 24\cdot 139^{3} + 117\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 47 + 71\cdot 139 + 15\cdot 139^{2} + 39\cdot 139^{3} + 128\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 65 + 102\cdot 139 + 123\cdot 139^{2} + 35\cdot 139^{3} + 74\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 82 + 8\cdot 139 + 18\cdot 139^{2} + 108\cdot 139^{3} + 100\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 107 + 49\cdot 139 + 102\cdot 139^{2} + 13\cdot 139^{3} + 64\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 120 + 108\cdot 139 + 53\cdot 139^{2} + 91\cdot 139^{3} + 91\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 126 + 37\cdot 139 + 83\cdot 139^{2} + 3\cdot 139^{3} + 7\cdot 139^{4} +O\left(139^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 128 + 95\cdot 139 + 67\cdot 139^{2} + 100\cdot 139^{3} + 111\cdot 139^{4} +O\left(139^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,3)(2,4)(5,7)(6,8)$$-2$
$2$$4$$(1,7,3,5)(2,8,4,6)$$0$
$2$$4$$(1,4,3,2)(5,6,7,8)$$0$
$2$$4$$(1,8,3,6)(2,5,4,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.