Properties

Label 2.28400.4t3.e.a
Dimension $2$
Group $D_{4}$
Conductor $28400$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{4}$
Conductor: \(28400\)\(\medspace = 2^{4} \cdot 5^{2} \cdot 71 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 4.2.142000.2
Galois orbit size: $1$
Smallest permutation container: $D_{4}$
Parity: odd
Determinant: 1.71.2t1.a.a
Projective image: $C_2^2$
Projective field: \(\Q(\sqrt{5}, \sqrt{-71})\)

Defining polynomial

$f(x)$$=$\(x^{4} - 5 x^{2} - 355\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 6.

Roots:
$r_{ 1 }$ $=$ \( 3 + 2\cdot 29 + 3\cdot 29^{2} + 10\cdot 29^{3} + 23\cdot 29^{5} +O(29^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 19\cdot 29 + 19\cdot 29^{2} + 23\cdot 29^{3} + 13\cdot 29^{4} + 24\cdot 29^{5} +O(29^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 24 + 9\cdot 29 + 9\cdot 29^{2} + 5\cdot 29^{3} + 15\cdot 29^{4} + 4\cdot 29^{5} +O(29^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 26 + 26\cdot 29 + 25\cdot 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} + 5\cdot 29^{5} +O(29^{6})\)  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.