Properties

Label 2.3e2_5e2_7e2.8t5.1c1
Dimension 2
Group $Q_8$
Conductor $ 3^{2} \cdot 5^{2} \cdot 7^{2}$
Root number 1
Frobenius-Schur indicator -1

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Basic invariants

Dimension:$2$
Group:$Q_8$
Conductor:$11025= 3^{2} \cdot 5^{2} \cdot 7^{2} $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 34 x^{6} + 29 x^{5} + 361 x^{4} - 305 x^{3} - 1090 x^{2} + 1345 x - 395 $ 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 $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 42\cdot 79 + 27\cdot 79^{2} + 30\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 63\cdot 79 + 39\cdot 79^{2} + 76\cdot 79^{3} + 39\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 + 29\cdot 79 + 38\cdot 79^{2} + 5\cdot 79^{3} + 26\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 45 + 41\cdot 79 + 21\cdot 79^{2} + 49\cdot 79^{3} + 25\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 49 + 16\cdot 79 + 55\cdot 79^{2} + 15\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 53 + 79 + 12\cdot 79^{2} + 68\cdot 79^{3} + 15\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 66 + 72\cdot 79 + 42\cdot 79^{2} + 11\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 72 + 48\cdot 79 + 78\cdot 79^{2} + 58\cdot 79^{3} + 40\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

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