Properties

Label 2.5e2_29e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 5^{2} \cdot 29^{2}$
Root number 1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$21025= 5^{2} \cdot 29^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 47 x^{6} + 40 x^{5} + 581 x^{4} - 220 x^{3} - 2038 x^{2} - 932 x - 109 $ 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 $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2 + 7\cdot 59 + 51\cdot 59^{2} + 13\cdot 59^{3} + 4\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 + 27\cdot 59 + 36\cdot 59^{2} + 42\cdot 59^{3} + 34\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 + 38\cdot 59 + 40\cdot 59^{2} + 43\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 9 + 55\cdot 59 + 59^{2} + 31\cdot 59^{3} + 42\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 12 + 30\cdot 59 + 19\cdot 59^{2} + 48\cdot 59^{3} + 3\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 19 + 23\cdot 59 + 57\cdot 59^{2} + 42\cdot 59^{3} + 43\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 28 + 42\cdot 59 + 39\cdot 59^{2} + 35\cdot 59^{3} + 41\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 42 + 12\cdot 59 + 48\cdot 59^{2} + 36\cdot 59^{3} + 15\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

Character values on conjugacy classes

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