Properties

Label 2.772.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $772$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(772\)\(\medspace = 2^{2} \cdot 193 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 4.2.148996.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.772.2t1.a.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(i, \sqrt{193})\)

Defining polynomial

$f(x)$$=$ \( x^{4} - 7x^{2} - 36 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 197 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 65 + 57\cdot 197 + 25\cdot 197^{2} + 181\cdot 197^{3} + 64\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 71 + 132\cdot 197 + 158\cdot 197^{2} + 46\cdot 197^{3} + 195\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 126 + 64\cdot 197 + 38\cdot 197^{2} + 150\cdot 197^{3} + 197^{4} +O(197^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 132 + 139\cdot 197 + 171\cdot 197^{2} + 15\cdot 197^{3} + 132\cdot 197^{4} +O(197^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

Cycle notation
$(1,2)(3,4)$
$(2,3)$

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.