Properties

Label 2.400.4t3.b.a
Dimension $2$
Group $D_{4}$
Conductor $400$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(400\)\(\medspace = 2^{4} \cdot 5^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.2000.1
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.4.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(i, \sqrt{5})\)

Defining polynomial

$f(x)$$=$\(x^{4} - 5\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.

Roots:
$r_{ 1 }$ $=$ \( 34 + 50\cdot 101^{2} + 68\cdot 101^{3} + 34\cdot 101^{4} +O(101^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 37 + 72\cdot 101 + 73\cdot 101^{2} + 57\cdot 101^{3} + 77\cdot 101^{4} +O(101^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 64 + 28\cdot 101 + 27\cdot 101^{2} + 43\cdot 101^{3} + 23\cdot 101^{4} +O(101^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 67 + 100\cdot 101 + 50\cdot 101^{2} + 32\cdot 101^{3} + 66\cdot 101^{4} +O(101^{5})\)  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.