Properties

Label 2.55.4t3.b.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 4.0.605.1 defined by $f= x^{4} - x^{3} + x^{2} + 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:
$r_{ 1 }$ $=$ $ 5 + 15\cdot 59 + 32\cdot 59^{2} + 37\cdot 59^{3} + 21\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 32 + 47\cdot 59 + 19\cdot 59^{2} + 34\cdot 59^{3} + 20\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 35 + 7\cdot 59 + 38\cdot 59^{2} + 51\cdot 59^{3} + 44\cdot 59^{4} +O\left(59^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 47 + 47\cdot 59 + 27\cdot 59^{2} + 53\cdot 59^{3} + 30\cdot 59^{4} +O\left(59^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,4)$
$(1,2)(3,4)$

Character values on conjugacy classes

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