Properties

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

Related objects

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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
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{5}, \sqrt{241})\)

Galois action

Roots of defining polynomial

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 values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,4)(2,3)$ $-2$
$2$ $2$ $(1,2)(3,4)$ $0$
$2$ $2$ $(1,4)$ $0$
$2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.