Properties

Label 2.68.4t3.c.a
Dimension 2
Group $D_4$
Conductor $ 2^{2} \cdot 17 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$68= 2^{2} \cdot 17 $
Artin number field: Splitting field of 8.0.21381376.2 defined by $f= x^{8} + 5 x^{6} + 4 x^{4} + 5 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.68.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{17})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 8 + 48\cdot 53 + 9\cdot 53^{2} + 48\cdot 53^{3} + 49\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 11 + 30\cdot 53 + 50\cdot 53^{2} + 12\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 32\cdot 53 + 41\cdot 53^{2} + 37\cdot 53^{3} + 45\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 24 + 16\cdot 53 + 24\cdot 53^{2} + 25\cdot 53^{3} + 50\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 29 + 36\cdot 53 + 28\cdot 53^{2} + 27\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 33 + 20\cdot 53 + 11\cdot 53^{2} + 15\cdot 53^{3} + 7\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 42 + 22\cdot 53 + 2\cdot 53^{2} + 40\cdot 53^{3} + 51\cdot 53^{4} +O\left(53^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 45 + 4\cdot 53 + 43\cdot 53^{2} + 4\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$

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

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

Character values on conjugacy classes

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