Properties

Label 2.2520.4t3.e.a
Dimension $2$
Group $D_{4}$
Conductor $2520$
Root number $1$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.0.60480.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.280.2t1.b.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{-6}, \sqrt{-70})\)

Defining polynomial

$f(x)$$=$\(x^{4} - 2 x^{3} + 9 x^{2} - 2 x + 1\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 53 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 4 + 38\cdot 53 + 3\cdot 53^{2} + 18\cdot 53^{3} + 18\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 28 + 24\cdot 53 + 15\cdot 53^{2} + 7\cdot 53^{3} + 52\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 36 + 12\cdot 53 + 51\cdot 53^{2} + 51\cdot 53^{3} + 16\cdot 53^{4} +O(53^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 40 + 30\cdot 53 + 35\cdot 53^{2} + 28\cdot 53^{3} + 18\cdot 53^{4} +O(53^{5})\)  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.