Properties

Label 2.2e3_41.4t3.4c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{3} \cdot 41 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$328= 2^{3} \cdot 41 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 3 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e3_41.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 83 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 13 + 82\cdot 83 + 9\cdot 83^{2} + 30\cdot 83^{3} + 74\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 23 + 76\cdot 83 + 33\cdot 83^{2} + 16\cdot 83^{3} + 44\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 61 + 6\cdot 83 + 49\cdot 83^{2} + 66\cdot 83^{3} + 38\cdot 83^{4} +O\left(83^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 71 + 73\cdot 83^{2} + 52\cdot 83^{3} + 8\cdot 83^{4} +O\left(83^{ 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.