Properties

Label 2.3e2_19.4t3.2c1
Dimension 2
Group $D_{4}$
Conductor $ 3^{2} \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$D_{4}$
Conductor:$171= 3^{2} \cdot 19 $
Artin number field: Splitting field of $f= x^{4} - x^{3} - 3 x^{2} + 2 x + 4 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 18 + 2\cdot 43 + 23\cdot 43^{2} + 25\cdot 43^{3} + 35\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 19 + 7\cdot 43 + 43^{2} + 38\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 12\cdot 43 + 7\cdot 43^{2} + 39\cdot 43^{3} + 2\cdot 43^{4} +O\left(43^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 30 + 20\cdot 43 + 11\cdot 43^{2} + 26\cdot 43^{3} + 33\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

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.