Properties

Label 2.5e2_41e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 5^{2} \cdot 41^{2}$
Root number -1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$42025= 5^{2} \cdot 41^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} - 63 x^{6} + 90 x^{5} + 1311 x^{4} - 20 x^{3} - 7702 x^{2} - 5524 x - 1009 $ 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 $\Q_{ 251 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 31 + 114\cdot 251^{2} + 222\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 56 + 6\cdot 251 + 134\cdot 251^{2} + 68\cdot 251^{3} + 186\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 58 + 181\cdot 251 + 246\cdot 251^{2} + 82\cdot 251^{3} + 47\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 82 + 29\cdot 251 + 179\cdot 251^{2} + 198\cdot 251^{3} + 231\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 135 + 142\cdot 251 + 64\cdot 251^{2} + 167\cdot 251^{3} + 156\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 174 + 30\cdot 251 + 117\cdot 251^{2} + 189\cdot 251^{3} + 73\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 225 + 239\cdot 251 + 212\cdot 251^{2} + 77\cdot 251^{3} + 239\cdot 251^{4} +O\left(251^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 246 + 122\cdot 251 + 186\cdot 251^{2} + 247\cdot 251^{3} + 245\cdot 251^{4} +O\left(251^{ 5 }\right)$

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

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

Character values on conjugacy classes

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