Properties

Label 2.776.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $776$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(776\)\(\medspace = 2^{3} \cdot 97 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.0.75272.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: even
Determinant: 1.776.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{2}, \sqrt{97})\)

Defining polynomial

$f(x)$$=$ \( x^{4} + 13x^{2} + 18 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 14 + 24\cdot 31 + 5\cdot 31^{2} + 2\cdot 31^{3} + 28\cdot 31^{4} + 2\cdot 31^{5} +O(31^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 + 4\cdot 31 + 14\cdot 31^{2} + 12\cdot 31^{3} + 14\cdot 31^{4} + 30\cdot 31^{5} +O(31^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 16 + 26\cdot 31 + 16\cdot 31^{2} + 18\cdot 31^{3} + 16\cdot 31^{4} +O(31^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 6\cdot 31 + 25\cdot 31^{2} + 28\cdot 31^{3} + 2\cdot 31^{4} + 28\cdot 31^{5} +O(31^{6})\) 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.