Properties

Label 2.17545.4t3.b.a
Dimension 2
Group $D_{4}$
Conductor $ 5 \cdot 11^{2} \cdot 29 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$17545= 5 \cdot 11^{2} \cdot 29 $
Artin number field: Splitting field of 4.0.87725.1 defined by $f= x^{4} - x^{3} + 30 x^{2} - 8 x + 229 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Even
Determinant: 1.145.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{29})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 13 + 73\cdot 151 + 126\cdot 151^{2} + 149\cdot 151^{3} + 56\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 + 97\cdot 151 + 83\cdot 151^{2} + 127\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 111 + 132\cdot 151 + 10\cdot 151^{2} + 97\cdot 151^{3} + 135\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 145 + 149\cdot 151 + 80\cdot 151^{2} + 78\cdot 151^{3} + 84\cdot 151^{4} +O\left(151^{ 5 }\right)$

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

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

Character values on conjugacy classes

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