Properties

Label 2.55.4t3.a.a
Dimension 2
Group $D_4$
Conductor $ 5 \cdot 11 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$55= 5 \cdot 11 $
Artin number field: Splitting field of 8.0.9150625.1 defined by $f= x^{8} - 3 x^{7} + 9 x^{6} - 13 x^{5} + 18 x^{4} - 11 x^{3} + 11 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.55.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{-11})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 11 + 56\cdot 59 + 31\cdot 59^{2} + 6\cdot 59^{3} + 27\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 13 + 35\cdot 59 + 54\cdot 59^{2} + 48\cdot 59^{3} + 58\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 14 + 54\cdot 59 + 47\cdot 59^{2} + 46\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 15 + 35\cdot 59 + 43\cdot 59^{2} + 34\cdot 59^{3} + 38\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 18 + 24\cdot 59 + 44\cdot 59^{2} + 30\cdot 59^{3} + 19\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 28 + 14\cdot 59 + 18\cdot 59^{2} + 46\cdot 59^{3} + 37\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 37 + 19\cdot 59 + 41\cdot 59^{2} + 49\cdot 59^{3} + 17\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 44 + 55\cdot 59 + 12\cdot 59^{2} + 18\cdot 59^{3} + 49\cdot 59^{4} +O\left(59^{ 5 }\right)$

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

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

Character values on conjugacy classes

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