Properties

Label 2.768.4t3.f.a
Dimension 2
Group $D_4$
Conductor $ 2^{8} \cdot 3 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$768= 2^{8} \cdot 3 $
Artin number field: Splitting field of 8.0.339738624.10 defined by $f= x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.3.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{-3})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 2 + 4\cdot 67 + 27\cdot 67^{2} + 18\cdot 67^{3} + 3\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 9 + 64\cdot 67 + 67^{2} + 47\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 12 + 31\cdot 67 + 26\cdot 67^{2} + 12\cdot 67^{3} + 24\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 13 + 19\cdot 67 + 3\cdot 67^{2} + 25\cdot 67^{3} + 4\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 54 + 47\cdot 67 + 63\cdot 67^{2} + 41\cdot 67^{3} + 62\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 55 + 35\cdot 67 + 40\cdot 67^{2} + 54\cdot 67^{3} + 42\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 58 + 2\cdot 67 + 65\cdot 67^{2} + 19\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 65 + 62\cdot 67 + 39\cdot 67^{2} + 48\cdot 67^{3} + 63\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

Character values on conjugacy classes

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