Properties

Label 2.21243.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $21243$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(21243\)\(\medspace = 3 \cdot 73 \cdot 97 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.2060571.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.21243.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{97}, \sqrt{-219})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 2x^{3} - 18x^{2} + 19x - 128 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 13 + 31\cdot 47 + 28\cdot 47^{2} + 36\cdot 47^{3} + 26\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 20 + 16\cdot 47 + 26\cdot 47^{2} + 36\cdot 47^{3} + 36\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 28 + 30\cdot 47 + 20\cdot 47^{2} + 10\cdot 47^{3} + 10\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 35 + 15\cdot 47 + 18\cdot 47^{2} + 10\cdot 47^{3} + 20\cdot 47^{4} +O(47^{5})\) Copy content Toggle raw display

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.