Properties

Label 2.3_7_13e2_19.4t3.1c1
Dimension 2
Group $D_{4}$
Conductor $ 3 \cdot 7 \cdot 13^{2} \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$67431= 3 \cdot 7 \cdot 13^{2} \cdot 19 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 52 x^{2} - 190 x - 209 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.3_7_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 18 + 63\cdot 97 + 86\cdot 97^{2} + 15\cdot 97^{3} + 78\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 34 + 78\cdot 97 + 19\cdot 97^{2} + 41\cdot 97^{3} + 31\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 67 + 82\cdot 97 + 75\cdot 97^{2} + 24\cdot 97^{3} + 53\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 76 + 66\cdot 97 + 11\cdot 97^{2} + 15\cdot 97^{3} + 31\cdot 97^{4} +O\left(97^{ 5 }\right)$

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.