Properties

Label 2.880.4t3.d.a
Dimension $2$
Group $D_{4}$
Conductor $880$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(880\)\(\medspace = 2^{4} \cdot 5 \cdot 11 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.4.4400.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.220.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{11})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 7x^{2} + 11 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 43 + 86\cdot 131 + 88\cdot 131^{2} + 100\cdot 131^{3} + 114\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 56 + 81\cdot 131 + 126\cdot 131^{2} + 42\cdot 131^{3} + 112\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 75 + 49\cdot 131 + 4\cdot 131^{2} + 88\cdot 131^{3} + 18\cdot 131^{4} +O(131^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 88 + 44\cdot 131 + 42\cdot 131^{2} + 30\cdot 131^{3} + 16\cdot 131^{4} +O(131^{5})\) 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.