Properties

Label 2.5e2_101e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 5^{2} \cdot 101^{2}$
Root number 1
Frobenius-Schur indicator -1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$255025= 5^{2} \cdot 101^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 164 x^{6} + 139 x^{5} + 6881 x^{4} - 11125 x^{3} - 63850 x^{2} + 149875 x - 56125 $ 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 $\Q_{ 71 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 1 + 29\cdot 71 + 47\cdot 71^{2} + 69\cdot 71^{3} + 27\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 10 + 51\cdot 71 + 38\cdot 71^{2} + 21\cdot 71^{3} + 67\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 17 + 43\cdot 71 + 19\cdot 71^{2} + 40\cdot 71^{3} + 47\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 25 + 70\cdot 71 + 14\cdot 71^{2} + 41\cdot 71^{3} + 22\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 28 + 2\cdot 71 + 11\cdot 71^{2} + 46\cdot 71^{3} + 48\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 40 + 61\cdot 71 + 23\cdot 71^{2} + 32\cdot 71^{3} + 43\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 41 + 8\cdot 71 + 54\cdot 71^{2} + 49\cdot 71^{3} + 39\cdot 71^{4} +O\left(71^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 52 + 17\cdot 71 + 3\cdot 71^{2} + 54\cdot 71^{3} + 57\cdot 71^{4} +O\left(71^{ 5 }\right)$

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

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

Character values on conjugacy classes

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