Properties

Label 2.2245.4t3.c.a
Dimension $2$
Group $D_{4}$
Conductor $2245$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(2245\)\(\medspace = 5 \cdot 449 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.1008005.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.2245.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{449})\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 23x^{2} + 20 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 1 + 3\cdot 11 + 9\cdot 11^{3} + 10\cdot 11^{4} + 6\cdot 11^{5} + 10\cdot 11^{6} + 11^{7} +O(11^{8})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 4\cdot 11 + 10\cdot 11^{2} + 8\cdot 11^{3} + 2\cdot 11^{4} + 6\cdot 11^{5} + 4\cdot 11^{6} + 10\cdot 11^{7} +O(11^{8})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 8 + 6\cdot 11 + 2\cdot 11^{3} + 8\cdot 11^{4} + 4\cdot 11^{5} + 6\cdot 11^{6} +O(11^{8})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 + 7\cdot 11 + 10\cdot 11^{2} + 11^{3} + 4\cdot 11^{5} + 9\cdot 11^{7} +O(11^{8})\) 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.