Properties

Label 2.3_61.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 3 \cdot 61 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$183= 3 \cdot 61 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} - 2 x^{2} + 3 x + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.3_61.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 13 + 42\cdot 73 + 42\cdot 73^{2} + 29\cdot 73^{3} + 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 26 + 25\cdot 73 + 35\cdot 73^{2} + 14\cdot 73^{3} + 15\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 48 + 47\cdot 73 + 37\cdot 73^{2} + 58\cdot 73^{3} + 57\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 61 + 30\cdot 73 + 30\cdot 73^{2} + 43\cdot 73^{3} + 71\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

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.