Properties

Label 2.2e2_5_13.4t3.4c1
Dimension 2
Group $D_{4}$
Conductor $ 2^{2} \cdot 5 \cdot 13 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$260= 2^{2} \cdot 5 \cdot 13 $
Artin number field: Splitting field of $f= x^{4} - 3 x^{2} - 2 x + 5 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.2e2_5_13.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots: \[ \begin{aligned} r_{ 1 } &= 9 + 11\cdot 97 + 23\cdot 97^{2} + 83\cdot 97^{3} + 59\cdot 97^{4} +O\left(97^{ 5 }\right) \\ r_{ 2 } &= 13 + 44\cdot 97 + 62\cdot 97^{2} + 38\cdot 97^{3} + 33\cdot 97^{4} +O\left(97^{ 5 }\right) \\ r_{ 3 } &= 29 + 23\cdot 97 + 65\cdot 97^{2} + 50\cdot 97^{3} + 3\cdot 97^{4} +O\left(97^{ 5 }\right) \\ r_{ 4 } &= 46 + 18\cdot 97 + 43\cdot 97^{2} + 21\cdot 97^{3} +O\left(97^{ 5 }\right) \\ \end{aligned}\]

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

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

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.