Properties

Label 2.299.4t3.a.a
Dimension 2
Group $D_{4}$
Conductor $ 13 \cdot 23 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$299= 13 \cdot 23 $
Artin number field: Splitting field of 4.2.3887.1 defined by $f= x^{4} - x^{3} + 5 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.299.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{13}, \sqrt{-23})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 5 + 53\cdot 131 + 74\cdot 131^{2} + 98\cdot 131^{3} + 15\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 67 + 42\cdot 131 + 115\cdot 131^{2} + 96\cdot 131^{3} + 42\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 91 + 60\cdot 131 + 47\cdot 131^{2} + 76\cdot 131^{3} + 12\cdot 131^{4} +O\left(131^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 100 + 105\cdot 131 + 24\cdot 131^{2} + 121\cdot 131^{3} + 59\cdot 131^{4} +O\left(131^{ 5 }\right)$

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

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

Character values on conjugacy classes

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