Properties

Label 2.128.4t3.b.a
Dimension 2
Group $D_4$
Conductor $ 2^{7}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$128= 2^{7} $
Artin number field: Splitting field of 8.0.4194304.1 defined by $f= x^{8} + 6 x^{4} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.8.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\zeta_{8})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11 + 19\cdot 41 + 9\cdot 41^{2} + 18\cdot 41^{3} + 15\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 + 32\cdot 41 + 15\cdot 41^{2} + 29\cdot 41^{3} + 14\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 15 + 11\cdot 41 + 20\cdot 41^{2} + 35\cdot 41^{3} + 35\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 17 + 26\cdot 41 + 17\cdot 41^{2} + 39\cdot 41^{3} +O\left(41^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 24 + 14\cdot 41 + 23\cdot 41^{2} + 41^{3} + 40\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 26 + 29\cdot 41 + 20\cdot 41^{2} + 5\cdot 41^{3} + 5\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 29 + 8\cdot 41 + 25\cdot 41^{2} + 11\cdot 41^{3} + 26\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 30 + 21\cdot 41 + 31\cdot 41^{2} + 22\cdot 41^{3} + 25\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

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