Properties

Label 2.2e3_3_5.8t11.2c2
Dimension 2
Group $Q_8:C_2$
Conductor $ 2^{3} \cdot 3 \cdot 5 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$Q_8:C_2$
Conductor:$120= 2^{3} \cdot 3 \cdot 5 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 3 x^{6} + x^{5} - 2 x^{4} - 3 x^{3} + 7 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $Q_8:C_2$
Parity: Odd
Determinant: 1.2e3_3_5.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 7 + 19\cdot 79 + 44\cdot 79^{2} + 40\cdot 79^{3} + 47\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 17 + 74\cdot 79 + 66\cdot 79^{2} + 61\cdot 79^{3} + 63\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 19 + 78\cdot 79 + 12\cdot 79^{2} + 15\cdot 79^{3} + 56\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 32 + 66\cdot 79 + 21\cdot 79^{2} + 70\cdot 79^{3} + 68\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 33 + 44\cdot 79 + 78\cdot 79^{2} + 65\cdot 79^{3} + 37\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 38 + 72\cdot 79 + 68\cdot 79^{2} + 13\cdot 79^{3} + 12\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 40 + 45\cdot 79 + 63\cdot 79^{2} + 67\cdot 79^{3} + 17\cdot 79^{4} +O\left(79^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 54 + 73\cdot 79 + 37\cdot 79^{2} + 59\cdot 79^{3} + 11\cdot 79^{4} +O\left(79^{ 5 }\right)$

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

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

Character values on conjugacy classes

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