Properties

Label 2.1205.4t3.a.a
Dimension $2$
Group $D_{4}$
Conductor $1205$
Root number $1$
Indicator $1$

Related objects

Learn more about

Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(1205\)\(\medspace = 5 \cdot 241 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 4.0.290405.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.1205.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{241})\)

Defining polynomial

$f(x)$$=$$ x^{4} - 2 x^{3} + 18 x^{2} - 17 x + 12 $.

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ $ 8 + 8\cdot 41 + 11\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 12 + 32\cdot 41 + 7\cdot 41^{2} + 36\cdot 41^{3} + 9\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 30 + 8\cdot 41 + 33\cdot 41^{2} + 4\cdot 41^{3} + 31\cdot 41^{4} +O\left(41^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 34 + 32\cdot 41 + 29\cdot 41^{2} + 38\cdot 41^{3} + 32\cdot 41^{4} +O\left(41^{ 5 }\right)$

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.